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Roquette-HVT1 - Fields Institute Communications Volume 00,...

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Unformatted text preview: Fields Institute Communications Volume 00, 0000 History of Valuation Theory Part I Peter Roquette Mathematisches Institut Universitat Heidelberg (Germany) roquette@uni-hd.de Abstract. The theory of valuations was started in 1912 by the Hungarian mathematician Josef Kurschak who formulated the valuation axioms as we are used today. The main motivation was to provide a solid foundation for the theory of p-adic elds as de ned by Kurt Hensel. In the following decades we can observe a quick development of valuation theory, triggered mainly by the discovery that much of algebraic number theory could be better understood by using valuation theoretic notions and methods. An outstanding gure in this development was Helmut Hasse. Independent of the application to number theory, there were essential contributions to valuation theory given by Alexander Ostrowski, published 1934. About the same time Wolfgang Krull gave a more general, universal de nition of valuation which turned out to be applicable also in many other mathematical disciplines such as algebraic geometry or functional analysis, thus opening a new era of valuation theory. In the present article which is planned as the rst part of more to come, we report on the development of valuation theory until the ideas of Krull about general valuations of arbitrary rank took roots. That is, we cover the pre-Krull era. As our sources we use not only the published articles but also the information contained in letters and other material from that time, mostly but not exclusively from the legacy of Hasse at the University library at Gottingen. Contents 1. Introduction 2. The beginning 2.1. Kurschak 2.2. Ostrowski 2.2.1. Solving Kurschak's Question 2.2.2. Revision: Non-archimedean Valuations 2 4 4 9 10 10 1991 Mathematics Subject Classi cation. Primary 12-03; Secondary 01A60. c 0000 American Mathematical Society 1 2 Peter Roquette 2.2.3. Ostrowski's Theorems 2.3. Rychl k and Hensel's Lemma 3. Valuations in Number Theory 3.1. Hasse: The Local-Global Principle 3.1.1. The motivation 3.1.2. The Local-Global Principle 3.2. Multiplicative structure 3.2.1. The cradle of local class eld theory 3.2.2. Hensel: The basis theorem 3.3. Remarks on p-adic analysis 3.4. Valuations on skew elds 3.4.1. Hasse's theorems 3.4.2. Consequences 3.5. Artin-Whaples: Axioms for Global Number Theory 4. Building the foundations 4.1. Dedekind-Hilbert theory 4.1.1. Krull, Deuring 4.1.2. Herbrand 4.2. F. K. Schmidt: Uniqueness theorem for complete elds 4.2.1. The Approximation Theorem 4.3. Structure of complete discrete elds. 4.3.1. The approach of Hasse and F. K. Schmidt 4.3.2. The Teichmuller character 4.3.3. Witt vectors 4.3.4. Mac Lane 5. Ostrowski's second contribution 5.1. Part I. Henselian elds. 5.1.1. Newton diagrams 5.2. Part II: Rami cation and defect 5.3. Part III: The general valuation problem 5.3.1. Pseudo-Cauchy sequences 5.3.2. Remarks on valuations of rational function elds 5.3.3. The general valuation problem 5.3.4. The Riemann surface of a eld References 1 Introduction 11 13 16 17 17 18 22 22 23 26 27 28 29 31 32 32 33 35 36 38 39 40 44 45 47 50 52 54 54 55 55 57 58 59 60 The origin of this paper was the manuscript for my lecture delivered at the Valuation Theory Conference in Saskatoon, August 1999. For publication in the proceedings volume I had intended to enlarge that manuscript such as to contain a survey of the complete history of valuation theory until this day. The importance of valuation theory in various applications is not questioned any more, and it would have been an exciting story to report on the impact which valuation theoretic notions and results have had in those applications. However, it soon turned out that for such a project I would have needed more pages History of Valuation Theory, Part I 3 as were allowed here and, what is more important, I would also have needed much more time for preparation. Well, here is at least the rst part of my intended \History of Valuation Theory", the next part(s) to follow in due course. This rst part covers the period of valuation theory which starts with Kurschak's de ning paper 1912] and ends about 1940, when the ideas of Krull about general valuations of arbitrary rank took roots, opening a new era of valuation theory. The discussion of Krull's seminal paper 1932g] itself will be included in the second part, as well as the application of Krull's valuation theory to other elds of mathematics, including real algebra, functional analysis, algebraic geometry and model theory. In other words: This rst part covers pre-Krull valuation theory. Let us make clear that \pre-Krull" is not meant to be understood in the sense of time, i.e., not \pre-1932". Rather, we mean those parts of valuation theory which do not refer to Krull's general concept of valuation of arbitrary rank and are based solely on Kurschak's notion of valuation, i.e., valuations of rank 1. 1 For a more detailed description of the content of this paper we refer to the table of contents. The reader will notice that one important part of valuation theory is not covered, namely the so-called non-archimedean analysis. Again, the reason is lack of space and time. For non-archimedean analysis the reader might consult Ullrich's paper 1995]. For the rich contributions of Krasner to valuation theory we refer to Ribenboim's article \Il mondo Krasneriano" 1985b]. Valuation theory has become important through its applications in many elds of mathematics. Accordingly the history of valuation theory has to take into account its applications. In the pre-Krull period (in the sense as explained above), it was the application to number theory which triggered much of the development of valuation theory. In fact, valuation theory was created primarily with the aim of understanding number theoretical concepts, namely Hensel's p-adic numbers for a prime number p . Although the formal de nition of valuation had been given by Kurschak 1912] it will appear that the ideas which governed valuation theory in its rst (pre-Krull) phase all came from Hensel. Thus Hensel may be called the father of valuation theory. Well, perhaps better \grandfather" because he never cared about the formal theory of valuations but only for his p-adic number elds. In any case, he has to be remembered as the great gure standing behind all of valuation theory (in the pre-Krull period). We have inserted a whole section devoted to the application of valuations to number theory. This exhibits the power of valuation theory and its role in the development of number theory. The reader will notice that in this section the name of Helmut Hasse will be dominant. Indeed, it was Hasse who successfully introduced and applied valuation theoretic ideas into number theory. He always propagated that the valuation theoretic point of view can be of help to better understand the arithmetic structure of number elds. 2 Van der Waerden, a witness of the times of the 1920s and 1930s, speaks of Hasse as \Hensel's best and great propagandist 1 This explains the fact that although here we do not discuss Krull's paper, we do discuss several papers which appeared after Krull's, i.e., after 1932. On the other hand, when in Part II we shall discuss Krull's paper 1932g], we shall have to consider a number of papers published before Krull's, belonging to strings of development leading to Krull's notion of general valuation, as for instance Hahn 1907a] and Baer 1927f] on ordered elds. 2 Perhaps it is not super uous to point out that this has nothing to do with what is called \Methodenreinheit ". 4 Peter Roquette of p-adic methods" 1975a]. If today the knowledge of valuations is considered a prerequisite to anyone who wishes to work in algebraic number theory then, to a high degree, this is due to Hasse's in uence. The application of valuation theory to algebraic geometry will be dealt with later, in the second part of this project. Besides of the original papers we have heavily used the information contained in letters and other material from that time. In those times mathematical information was usually exchanged by letters, mostly handwritten, before the actual publication of papers. Such letters often contain valuable information about the development of mathematical ideas. Also, they let us have a glimpse of the personalities behind those ideas. To a large degree we have used the legacy of Helmut Hasse contained in the University Library at Gottingen. Hasse often and freely exchanged mathematical ideas and informations with his correspondence partners; there are more than 6000 letters. Besides of the Gottingen library, we have also used material from the archive of Trinity College (Cambridge), and from the Ostrowski legacy which at present is in the hands of Professor Rita Jeltsch-Fricker in Kassel. Remark. The bibliography contains the papers which we have cited in this article. We have sorted it by the year of publication, in order to give the reader some idea of the progress in time, concerning the development of valuation theory. But as we all know, the publication date of a result is usually somewhat later than the actual date when it was discovered, or when it was communicated to other mathematicians. Thus the ordering of our bibliography gives only a rough picture of the actual development. Sometimes the letters which were exchanged give more precise information; if so then we have mentioned it in the text. Acknowledgement. I would like to thank F.-V. Kuhlmann for carefully reading the manuscript, and for his critical and thoughtful comments. 2 The beginning 2.1 Kurschak. Valuations have been around in mathematics since ancient times. When Euclid had proved the uniqueness of prime decomposition, then this result permitted to code the natural numbers by the exponents with which the various primes p occur in these numbers; those exponents in fact represent the padic valuations used in number theory. Similarly in the theory of functions: the order of a holomorphic function at a given point P on a Riemann surface represents a valuation on the respective function eld, and the function is uniquely determined, up to a constant factor, by its behavior at those valuations. Valuations of this kind have been exploited heavily in number theory and complex function theory during the 19th century. However, Valuation Theory as a separate and systematic mathematical research, based on a set of axioms, started in the 20th century only, in the year 1912 when the Hungarian mathematician Josef Kurschak (1864{1933) announced at the Cambridge International Congress of Mathematicians the rst abstract structure theorems on valued elds 1912]. The paper itself, written in German, appeared one year later in Crelle's Journal 1913]. At the beginning of the paper we nd the familiar four axioms for a History of Valuation Theory, Part I valuation: 5 kak > 0 if a 6= 0, and k0k = 0 k1 + ak 1 + kak kabk = kak kbk 9a : kak 6= 0; 1 (1) (2) (3) (4) Here, a and b range over the elements of a given eld K , and the values kak are supposed to be real numbers. Kurschak uses already the name \Bewertung " which is still used today, and which is translated into English as \valuation". He tells us that he had chosen this name in order to indicate that it is meant as a generalization of the notion of \absoluter Wert " (absolute value) which he understood as the ordinary absolute value de ned in the real or the complex number eld. 3 The main purpose of Kurschak's paper was to present a proof of the following theorem: Every valued eld K admits a valued eld extension C K which is algebraically closed and complete . Here, \complete" refers to the given valuation and signi es that every Cauchy sequence is convergent. Kurschak uses the German terminology \perfekt " for \complete". Kurschak's terminology became widely used in the twenties and early thirties but was later abandoned in favor of \komplett " or \vollstandig " in order to avoid misunderstandings with the English terminology. 4 Kurschak says explicitly that he was inspired by Hensel's book on algebraic numbers 1908]. His aim is to give a solid foundation of Hensel's p-adic algebraic numbers, in a similar way as Cantor had given for the real and complex numbers. Thus we see that the main motivation to introduce valuation theory came from algebraic number theory while the model for the axioms and for the method of reasoning was taken from analysis. Kurschak's paper may be viewed as one of the rst instances where \analytical algebra" or, as we prefer today, \topological algebra" was deliberately started. Today, we usually de ne the p-adic number eld Q p as the completion of the rational number eld Q with respect to its p-adic valuation, and C p as the completion of the algebraic closure of Q p , thereby using the fact that the valuation of the complete eld Q p extends uniquely to its algebraic closure. In so doing we follow precisely Kurschak's approach. Before Kurschak, Hensel had de ned p-adic algebraic numbers through their power series expansions with respect to a prime element. This procedure was quite unusual since Hensel's power series do not converge in the usual sense, and hence do not represent \numbers" in the sense as understood at the time, i.e., they are not complex numbers. Accordingly there was some widespread uneasiness about Hensel's p-adic number elds and there were doubts whether they really existed. Kurschak's paper was written to clear up this point. 3 In modern (Bourbaki) terminology, all Kurschak valuations are called \absolute values", and the word \valuation" refers to the more general notion as de ned later by Krull. In this article we use \valuation" in the sense of Kurschak. 4 In English, the property \perfect" of a eld signi es that the eld does not have proper inseparable algebraic extensions; in German this property is called \vollkommen ", as introduced by Steinitz 1910]. 6 Peter Roquette As to the choice of his axioms (1){(4), Kurschak refers to Hensel's article 1907] where Hensel, in the case of p-adic algebraic numbers, had already de ned some similar valuation function; the formal properties of that function are now used by Kurschak as his axioms. 5 Kurschak's proof of his main theorem proceeds in three steps: b Step 1.: Construction of the completion K of a valued eld K . 6 b to its algebraic closure. Step 2.: Extending the valuation from K Step 3.: Proving that the completion of an algebraically closed valued eld is algebraically closed again. In Step 1 he proceeds by means of what he calls Cantor's method, i.e., the b elements of K are de ned to be classes of Cauchy sequences modulo null sequences. \Cantor's method" seems to have been well known at that time already since Kurschak does not give any reference. 7 In Step 2, the existence of the algebraic closure of a eld is taken from Steinitz' great Crelle paper 1910] which established the fundamentals of elds and the structure of their extensions. In fact, Kurschak's paper rests heavily on the paper of Steinitz and can be regarded as a natural continuation thereof. 8 In his introduction he refers to Steinitz and says that he (Kurschak) will now introduce into eld theory a new notion, i.e., valuation. (\In dieser Abhandlung soll in die Korpertheorie ein neuer Begri eingefuhrt werden : : : ") In Step 3, the method of proof is copied from the method of Weierstra in 1891], where Weierstra gave a new proof that the complex number eld C is algebraically closed. That theorem was called, in those times, the \fundamental theorem of algebra". Accordingly, Kurschak now calls his theorem the \fundamental theorem of valuation theory". He points out that the arguments of Weierstra are valid for an arbitrary algebraically closed valued eld instead of the eld of all algebraic numbers { but nevertheless he has to add an extra discussion of inseparability in the case of characteristic p > 0 which, of course, does not appear in Weierstra ' paper. Kurschak does not always give detailed proofs; instead he says: Da meine Untersuchungen beinahe ausnahmlos nur selbstverstandliche Verallgemeinerungen bekannter Theorien sind, so scheint es mir zu genugen, wenn ich in den nachsten Kapiteln die einzelnen De nitionen und Satze ausfuhrlich darlege. Auf die Details der Beweise werde ich nur selten eingehen. 5 Hensel's de nition was erroneous because of a missing minus sign in the exponent; Kurschak corrects this politely by saying that he has replaced p by p?1 . 6 Concerning our notation: We do not follow necessarily the notation used by the authors of the papers discussed here. Instead, we try to use a uni ed notation for the convenience of the b reader. In particular, K is not the notation of Kurschak (he writes K 0 for the completion and calls it the \derived eld" of K ). Similarly, the notations Q , Q p , C etc. which are generally used today, were not yet in use at the time of the papers discussed in our article. 7 Compare Cantor's paper 1883], in particular pp. 567 . 8 Such a continuation of the Steinitz paper seems to have been in the spirit of Steinitz himself. For, Steinitz says in the introduction of his paper 1910], that his article concerns the foundations of eld theory only. He announces further investigations concerning the application of eld theory to geometry, number theory and theory of functions. But those further articles have never appeared, for reasons which are not known to us. We may speculate that the reason is to be found in Kurschak's publication which was followed by those of Ostrowski. History of Valuation Theory, Part I 7 My investigations are, almost without exception, straightforward generalizations of known theories, and hence it seems to be su cient that in the next chapters I present in detail the various de nitions and theorems only. The details of the proofs will be given only occasionally. One occasion for Kurschak to go into more detail of proof is in Step 2 when he discusses the possibility of extending the valuation from a complete eld K to its e algebraic closure K . Let be algebraic of degree n over K and N its reduced norm, i.e., the constant coe cient of the irreducible monic polynomial of over K . Then Kurschak gives the formula k k = kN k1=n (5) e which de nes an extension of the given valuation of K to its algebraic closure K . This formula had been given by Hensel in the case of the p-adic numbers. The main point in the proof is to derive the triangle inequality (2) from the de nition (5). If the given valuation is non-archimedean then, as Kurschak observes, the method developed by Hensel in his book 1908] for p-adic numbers is applicable. Kurschak does not use the word \non-archimedean" which came into use later only 9 ; but he says very clearly that Hensel's methods work: : : : wenn die Bewertung von der besonderen Bescha enheit ist, da ka + bk nicht gro er ist als die gro ere der Zahlen kak und kbk. : : : if the valuation has the special property that ka + bk is not greater than the greater of the numbers kak and kbk. More precisely, Kurschak says (without proof) that the following lemma, which today is called \Hensel's Lemma", is valid in every complete non-archimedean valued eld, and that Hensel's proof applies: Lemma: If f (x) = xn + an?1 xn?1 + + a0 is irreducible, and if ka0 k 1 then each coe cient kai k 1 (0 i n ? 1). Starting from the Lemma, Kurschak shows, the triangle inequality (2) is easily obtained as follows: In the above Lemma, take f (x) 2 K x] to be the monic irreducible polynomial for ; then a0 = N . The irreducible polynomial for 1+ is f (x ? 1) with the constant coe cient N (1 + ) = f (?1) = (?1)n + an?1 (?1)n?1 + ? a1 + a0 Using the Lemma and the fact that the given valuation on K is non-archimedean we see that ka0 k = kN k 1 =) kN (1 + )k 1 : This implies the triangle inequality (2) in the algebraic closure of the complete eld K. But Kurschak is looking for a uni ed proof which simultaneously covers all cases, archimedean and non-archimedean alike. He nds the appropriate method in the thesis of Hadamard 1892]. Hadamard had considered only the case of the 9 The terminology \non-archimedean" and \archimedean" for valuations is introduced in Ostrowski's paper 1917]. Today the terminology \ultrametric" for \non-archimedean" is widely used. 8 Peter Roquette complex eld C as base eld but Kurschak observes that his arguments are valid in an arbitrary complete valued eld K . Given a monic polynomial f (x) = xn + an?1 xn?1 + + a0 over the complete eld K , Hadamard's method permits to determine the radius of convergence rf of the power series n Pf (x?1 ) = fxx) = 1 + c1 x?1 + c2 x?2 + ( in the variable x?1 . If f (x) is irreducible over K then Kurschak is able to conclude that rf = ka0 k1=n = kN k1=n (6) where is a root of f (x). Now, the triangle inequality (2) follows by comparing the radii of convergence of Pf (x?1 ) and Pg (x?1 ) with g(x) = f (x ? 1). This discussion in Kurschak's paper seems to be somewhat lengthy. Obviously, he could not yet know Ostrowski's theorem 1918], which says that for an archimedean valuation, the only complete elds are R (the reals) and C (the complex numbers). In these two cases the problem of extending the valuation is trivial, and so Kurschak's paper could indeed have been considerably shortened by concentrating on the non-archimedean case and using Hensel's Lemma { as is the usual procedure today. Nevertheless it seems interesting that a uni ed proof, applicable in the archimedean as well as the non-archimedean case, does exist, a fact which today seems to be forgotten. It would be of interest to simplify Kurschak's proof using what today is known from analysis in complete valued elds, and thus give a simple uni ed treatment for the archimedean and the non-archimedean case. Hensel's Lemma can then be deduced from this. Concerning step 3, Kurschak wonders whether this step would really be necessary. Perhaps the algebraic closure of a complete eld is complete again? He doubts that this is the case for Q p but is not able to decide the question. But already in the same year, in the next volume of Crelle's Journal, the question will be settled in a paper by Ostrowski (see section 2.2). When Kurschak published his valuation theory paper in 1912 he was 48. As far as we were able to nd out, this was the last and only paper of Kurschak on valuation theory. 10 His list of publications comprises about 80 papers between the years 1887 and 1932, on a wide variety of subjects including analysis, calculus of variations and elementary geometry. He held a position at the Technical University in Budapest, and he became an in uential academic teacher, recognizing and assisting mathematical talents. 11 We have no knowledge of how Kurschak became interested in the subject of valuations. Had he been in contact with Kurt Hensel in Marburg? Or with some 10 In 1923e] he published a method to determine irreducibility of a polynomial over Q p by means of Newton's diagram method (which he calls the Puiseux diagram method). It is curious that Kurschak, the founder of valuation theory, restricted his investigation to polynomials with integer coe cients; he did not mention that the Newton diagram method can be applied without any extra e ort to polynomials over an arbitrary non-archimedean complete valued eld, as Ostrowski later 1934] observed. 11 E.g., John von Neumann was one of his students. { I am indebted to Kalman Gyory for providing me with biographical information about Kurschak. See also the books 1992] and 1996a] on Mathematics in Hungary. History of Valuation Theory, Part I 9 other colleague who knew Hensel, or with Steinitz? In any case, his paper appeared just at the right time, providing a solid base for the study of valued elds which started soon after. It seems that the axiomatization of the theory of Hensel's padic elds was overdue at that time, after the work of Hensel and Steinitz. Thus, if Kurschak had not written this paper then perhaps some other mathematician would have done it at about the same time and in a similar spirit. This of course does not diminish Kurschak's merits. His paper, like other publications of him, is very clearly written { certainly this contributed to the quick and wide distribution of his notions and results. In this connection it may be not without interest that about the same time, A. Fraenkel had already presented another axiomatic foundation of Hensel's p-adic number elds 1912a]. At that time Fraenkel stayed in Marburg with Hensel. His paper had appeared in Crelle's Journal in the volume before Kurschak's. Neither Fraenkel nor Kurschak cites the other, and so both papers seem to have been planned and written independently from each other. Fraenkel's paper is completely forgotten today. Fraenkel in his memoirs 1967] explains this in rather vague terms by saying that his axiomatization had been purely formal whereas Kurschak's takes into account the content of Hensel's theory (\inhaltliche Begrundung "). It is not clear what Fraenkel may have meant by this. In retrospective we see that Kurschak's paper, in contrast to Fraenkel's, opened up a new branch of mathematics, i.e., valuation theory, which turned out to be successfully applicable in many parts of mathematics. Let us quote a remark by Ostrowski from 1917]: Uberhaupt kann, wie uns scheint, ein vollstandiger Einblick in die Natur dieser merkwurdigen Bildungen der p-adischen Zahlen] nur vom allgemeinen Standpunkt der Bewertungstheorie gewonnen werden. Anyway, in our opinion, a complete understanding of those curious constructions the p-adic numbers] can only be obtained from the point of view of general valuation theory. Ostrowski does not mention Fraenkel in this connection, but since both had been in Marburg at that time it is well conceivable that they had discussed the question of an adequate foundation of Hensel's theory of p-adic numbers, in the light of the two papers by Fraenkel and by Kurschak. And the above remark represents Ostrowski's opinion on that question. 2.2 Ostrowski. After Kurschak had started the theory of valued elds it was Alexander Ostrowski (1893 { 1986) who took over and developed it further to a considerable degree. Ostrowski was born in Kiev and had come to Marburg in 1911, at the age of 18, in order to study with Hensel. Abraham Fraenkel, who had been in Marburg at that time, recalls in his memoirs 1967] that Ostrowski showed unusual talent and originality (\eine ungewohnliche Begabung und Originalitat "). From Fraenkel we also learn that Ostrowski had been advised by his professor in Kiev to go to Marburg. At that time the dominant mathematician in Kiev was D.A. Grave. 12 It seems remarkable to us that Grave sent his extraordinarily gifted student to Marburg and not to Gottingen, although the latter was worldwide known 12 A biography on Grave is contained in \The MacTutor History of Mathematics archive", at the internet address http://www-groups.dcs.st-and.ac.uk/ history/. 10 Peter Roquette for its inspiring mathematical atmosphere, and Landau in Gottingen had o ered to accept him as a student. The reason for this advice may have been that he (Grave) was well acquainted with the theory of Hensel's p-adic numbers and considered it to be something which would become important in the future. In any case, we know that p-adics were taught in Kiev. 13 In Marburg, the home of p-adics, Kurschak's paper was thoroughly studied and discussed. Soon the young Ostrowski found himself busily engaged in developing valuation theory along the tracks set by Kurschak. 2.2.1 Solving Kurschak's Question. In his rst 14 paper 1913a], published in Crelle's Journal, Ostrowski solved the open question of Kurschak mentioned in the foregoing section, namely whether the algebraic closure of a complete valued eld is complete again. Ostrowski proves that a separable 15 algebraic extension of a complete valued eld K is complete if and only if it is of nite degree over K . From this he concludes e e that the algebraic closure K of a complete eld K is complete if and only if K is e is separable over K or not. 16 In nite over K . This holds regardless of whether K particular it follows that the algebraic closure of Q p is not complete, which answers the question of Kurschak. Ostrowski could not know the Artin-Schreier theorem which was discovered e much later in 1927a] only. According to Artin-Schreier, the algebraic closure K of a eld K is almost never nite over K , the exceptions being the real closed elds { ~ and of course the trivial cases K = K . Moreover it can be shown that a real closed eld K is never complete with respect to a rank 1 valuation except when K = R . Hence Ostrowski's result implies that the algebraic closure of a complete eld K is almost never complete, with the only exception K = R { and of course the trivial cases when K itself is algebraically closed. Ostrowski was quite young when this paper appeared in 1913 (he was 20), and it is perhaps due to this fact that the paper appears somewhat long-winded. But Ostrowski continued to work on valuation theory. In his next papers he not only simpli ed the proofs but also produced a number of further fundamental results. 2.2.2 Revision: Non-archimedean Valuations. In his paper 1917], Ostrowski sets out to prove the results of his 1913 paper anew and, he says, with proofs much more elementary. 13 It is known that N. Chebotarev, who was a contemporary of Ostrowski and had studied in Kiev, was well acquainted with p-adic numbers. When Chebotarev met Ostrowski 1925 in Gottingen, the latter posed him a problem coming from complex analysis, and Chebotarev was able to solve it by means of p-adic numbers; see the article by Lenstra and Stevenhagen 1996]. 14 Actually, Ostrowski had already published another mathematical article, on nite elds, which had appeared (in Russian) in the S. B. Phys. Math. Ges. Kiew. See vol. 3 of Ostrowski's Collected Papers 1983]. 15 He does not use the word \separable" which was coined only later by van der Waerden. Instead he uses \of the rst kind" (von 1. Art ) as introduced by Steinitz 1910]. Ostrowski then de nes \of the second kind" (von 2. Art ) to mean \purely inseparable" in today's terminology. It seems somewhat curious that Steinitz himself, although he introduced the notion \of the rst kind", did not introduce the terminology \of the second kind" (von 2. Art ). This seems to be due to Ostrowski. A purely inseparable extension eld is called by Steinitz \root eld" (Wurzelkorper ). 16 For, if K is in nite over the complete eld K then he shows that the separable closure too e is in nite over K . By the way, this implies that a separably closed complete eld is algebraically closed { a result which later was rediscovered by F. K. Schmidt; see section 4.2. History of Valuation Theory, Part I 11 From the start he considers non-archimedean valuations only. This is permitted since in another paper 1918] he is able to classify the complete archimedean valued elds. (See section 2.2.3.) Here we nd the following results: (1) A new and simple proof of the fact that any nite extension of a complete eld is complete again, this time including the case of inseparable extensions. This proof has become standard today, and with the same arguments one usually shows that any normed vector space of nite dimension over a complete eld is complete again. By the way, in this proof the non-archimedean property of the valuation is not used, hence archimedean valuations are included. (2) An addition to Kurschak's result by showing that for a complete ground eld, the extension of the valuation to its algebraic closure is unique { a fact which, Ostrowski says, is implicitly contained also in Kurschak's construction but his own proof is now much easier. (3) A Lemma which later became known as \Krasner's Lemma". This is a very useful version of Hensel's Lemma and says the following: Let K be complete with respect to a non-archimedean valuation, and let be algebraic and separable over K . Denote by the minimal distance of to its conjugates over K . Then: Lemma: If L is any valued algebraic extension of K such that the distance of to L is < then 2 L. It is true that Ostrowski states the contention for =2 instead of but from his proof it is clear that indeed the lemma holds for because of the strong triangle inequality which Ostrowski does not use in this instance. 17 We see: Although the Lemma had been stated and used by Ostrowski in 1917] already, it has come to the attention of the mathematical community much later only, under the name of \Krasner's Lemma". This is another instance of a situation which we can observe in every corner of mathematics: The name of a mathematical result or notion does not always re ect the historical origin. (4) A simple proof, using the above Lemma, that every in nite separable algebraic extension of a complete eld is not complete. The same is proved for inseparable extensions if the inseparability exponent is in nite. From this Ostrowski concludes again his main result of his earlier paper 1913a]: that the algebraic e e closure K of a complete eld K is complete if and only if K is nite over K . 2.2.3 Ostrowski's Theorems. Although the paper 1918] appeared after 1917] it was completed already before 1917]. It is dated April 1916. The paper contains two fundamental theorems. In the rst theorem, all possible valuations of the rational number eld Q are determined. Ostrowski nds that, up to equivalence, these are precisely the known ones. Two valuations k k1 and k k2 of a eld K are said to be equivalent if one is a power of the other, i.e. kak2 = kakr for every a 2 K , where the exponent 1 r does not depend on a. It is clear that equivalent valuations generate the same 17 Ostrowski seems to claim that under the hypothesis of the Lemma, is the only one among its K -conjugates whose distance to L is < ; he uses the words \eine einzige ". Obviously this uniqueness assertion is false in general, there are trivial counterexamples. In his proof Ostrowski does not deal with this uniqueness, and when he applies the Lemma he does not use it. Hence we believe that this is not really a mathematical error, but it re ects a certain linguistic slip of the author. In a later paper 1934] he corrects it and says that \eine einzige " should be replaced by \eine solche ", which gives the formulation as we have stated it. { In general, though, Ostrowski's papers are very clear and sharp; in 1988] they are praised as \true pearls in mathematical literature" (wahre Perlen im mathematischen Schrifttum ). 12 Peter Roquette metric topology of the eld K and, hence, lead to isomorphic completions. Usually, equivalent valuations will not be distinguished. More precisely, an equivalence set of valuations will be called a prime of the eld, usually denoted by a symbol like b p, and the corresponding completion will be denoted by Kp or simply Kp . 18 Hence, what Ostrowski proves in 1918] can be expressed by saying that every prime p of the rational number eld Q either corresponds to a prime number p and thus belongs to the usual p-adic valuation of Hensel-Kurschak, or p = p1 is an in nite prime and belongs to the ordinary absolute value. Today this theorem belongs to the basics of a rst course in valuation theory. The usual proof presented today is due to E. Artin 1932f] who uses essentially the same ideas as Ostrowski but is able to streamline the proof into 2 pages. Perhaps it may be mentioned that Hasse was so delighted about Artin's version of Ostrowski's proof that he included it into his book \Zahlentheorie " 1949] with the comment: \Dieser schone Beweis geht auf Artin zuruck." (This beautiful proof is due to Artin.) Actually, the above de nition of a \prime" p in an arbitrary eld K is given by Ostrowski in the case of non-archimedean valuations only. In this case, he also switches to the additive notation of valuations by de ninig v(a) = ? log kak. 19 Of course it is a question of taste whether one wishes to include archimedean valuations or not among what one calls \primes". As Hasse 1927], x2 has pointed out, the experiences in class eld theory strongly indicate that archimedean valuations should be treated, as far as possible, on the same footing as non-archimedean ones, and this is generally accepted today. In any case, it seems worthwhile to state that Ostrowski was the rst one who formulated explicitly the notion of \prime" in any abstract eld by means of valuation theory. Below we shall see that later in his long paper 1934] he will enlarge on this, studying the set of all those primes which he calls the \Riemann surface" of the eld. The second main result in this paper consists of what today is simply called \Ostrowski's theorem" in valuation theory. It is concerned with archimedean valuations and asserts that the only elds which are complete with respect to an archimedean valuation are the elds R of real numbers and C of complex numbers, up to isomorphisms as topological elds . The topological isomorphism implies that the given valuation becomes equivalent to the ordinary absolute value. As said already in section 2.1, this theorem permits a substantial simpli cation in Kurschak's discussion of extending the valuation from a complete eld to an algebraic extension. For, in view of Ostrowski's theorem it is now possible to consider non-archimedean valuations only, and for these Hensel's Lemma can be used. Hence it will not be necessary any more to refer to Hadamard's results on the radius of convergence of certain power series. 20 18 In the literature there are various other de nitions of \primes" of a eld, all of which are equivalent. Also, instead of the terminology \prime", other names are used too, e.g. \prime divisors", \prime spots", \places" or \points" of a eld. For the purpose of this report we will take the de nition of \prime" as given in the text. If the valuations are archimedean then the corresponding prime is called an \in nite" prime and denoted by a symbol like 1 . 19 The minus sign is missing in his de nition, and this is corrected in 1934]. 20 Ostrowski points out that Hadamard's investigations can be regarded as a generalization of Bernoulli's method to approximate the roots of an algebraic equation. Ostrowski's proof, he says, uses another generalization of the same method { but only in the case of quadratic extensions which su ces for his theorem and therefore is much simpler. p p History of Valuation Theory, Part I 13 One of the rst readers of Ostrowski's paper was Emmy Noether. In a postcard to Ostrowski 21 she writes: Ihre Funktionalgleichungen habe ich angefangen zu lesen, und sie interessieren mich sehr. Kann man wohl den allgemeinsten Korper charakterisieren, der einem Teiler des Korpers aller reellen Zahlen isomorph ist ? : : : I have started to read your functional equations 22 and I am very interested in it. Is it perhaps possible to characterize the most general eld which is isomorphic to a sub eld 23 of the eld of all real numbers ? Emmy Noether does not only express her interest in Ostrowski's work but immediately poses the correct question: Which elds can be isomorphically embedded into R ? Her question was answered later in 1927a] by Artin-Schreier's theory of formally real elds. We have cited this postcard in order to put into evidence that Emmy Noether has shown interest in the development of valuation theory right from the beginning, although she herself was never active in this direction. Later in 1930/31 she actively participated, together with Richard Brauer and Helmut Hasse, in the proof of the Local-Global Principle for algebras; see section 3.4.2. 2.3 Rychl k and Hensel's Lemma. It was the Czech mathematician Karel Rychl k (1885{1968) who set out to present explicitly such simpli cation as mentioned by Ostrowski and Kurschak, for non-archimedean valuations only. His paper 1919] appeared in a Czech journal, soon after Ostrowski's paper 1918]. But since it was written in Czech language it seems that it was not properly noticed by the mathematical community (though it was refereed in the Jahrbuch uber die Fortschritte der Mathematik , vol. 47). Later in 1923 Rychl k published essentially the same paper in German language, in Crelle's Journal 1923f]. 24 This paper certainly was read and appreciated. Rychl k appears to have been strongly in uenced by Hensel's theory of p-adic numbers, through Hensel's various papers and particularly through Hensel's books 1908], 1913b] which he knew well. As a result of his studies he had published, since 1914, some earlier papers already presenting Hensel's ideas of the foundation of algebraic number theory. Those papers were also in Czech language. Of particular interest to us is the paper 1916a] where Rychl k, in a postscript, presented Kurschak's construction for the p-adic elds Q p . 25 It is apparent that Rychlik was 21 The postcard is not dated; it seems that it was written in early 1916, some months before Ostrowski submitted his manuscript 1918] to the Acta Mathematica. { I am indebted to Prof. Rita Jeltsch-Fricker to give me access to the legacy of Ostrowski where I found several postcards from Emmy Noether. 22 By \functional equations" she refers to the title of Ostrowski's paper which reads: Uber einige Losungen der Funktionalgleichung '(x) '(y) = '(xy). (On some solutions of the functional equation : : : ) 23 She writes \Teiler" which usually translates with \divisor". In the present context it obviously means \sub eld" in today's terminology. 24 Usually this paper is cited as having appeared in 1924. But volume 153 of Crelle's Journal consisted of two issues and the rst issue, containing Rychl k's paper, appeared on Aug 27, 1923 already. 25 More generally, Rychl k covered also the Hensel g-adic rings Q g for an arbitrary integer g > 1. In this case the ordinary, Kurschak's, notion of valuation is not adequate; instead he had to use what later was called a \pseudo-valuation"; this is a function which satis es all the conditions for a valuation except the multiplicative rule (3) which is to be replaced by kabk kakkbk : Thus 14 Peter Roquette not a newcomer in valuation theory when he wrote the paper 1919], or its German version 1923f]. 26 In these papers 1919], 1923f] under discussion, Rychl k cites Kurschak and Ostrowski and their results mentioned above. For a complete eld with a nonarchimedean valuation, he says, he is now going to present a simple proof for the prolongation of the valuation to an algebraic extension. 27 He will present in full detail the proof which Kurschak, by a sort of hand waving, had indicated but not explicitly presented. Seen from today, Rychl k is taking the relevant results and proofs from Hensel's book 1908] and repeating them in the more general framework of an arbitrary complete, non-archimedean valued eld. This seems rather trivial to us but we should keep in mind that at those times, valuation theory was quite new and people were not yet used to valuation theoretic arguments. Moreover, there was indeed some di erence between Hensel's arguments for p-adic elds and those for general complete elds. Namely, the valuation of a p-adic eld is discrete in the sense that its value group is a discrete subgroup of the positive reals; this implies that the maximal ideal of the valuation ring is generated by one element (\prime element"). The relevant approximation algorithm in Hensel's treatment proceeds sucessively with respect to the powers of a prime element. But in general the valuation will not be discrete and hence there does not exist a prime element. Therefore in Hensel's arguments, the powers of the prime element have to be replaced by the powers of some suitable element in the maximal ideal. For us, this seems quite natural and straightforward. As said above already, apparently this was not considered to be trivial in those times, when abstract valuation theory had just been started. Sometimes in the literature there appears what is called the \Hensel-Rychl k Lemma". This terminology recalls that Hensel was the rst to discover the validity and importance of the Lemma in the case of p-adic elds and their nite extensions, and that Rychl k did verify its validity for arbitrary complete non-archimedean valued elds. 28 Today, however, the name \Hensel's Lemma" has become standard, and the name \Hensel-Rychlik" is used sometimes only to distinguish a certain version of Hensel's Lemma from others. We see again, that such names do not re ect the historical background but are assigned somewhat arbitrarily by the mathematical community { as we have noticed already in the foregoing section with \Krasner's Lemma" which in fact is due to Ostrowski. Now what does the statement say which is known as \Hensel-Rychl k Lemma"? Rychl k preceded Mahler 1936]- 1936c], who introduced the formal notion of pseudo-valuation, by 20 years. 26 I am indebted to Dr. Magdalena Hyksova, and also to Professor Radan Kucera, for translating part of the Czech papers of Rychl k. Dr. Hyksova also informed me about many interesting biographical details; she is preparing a monograph (in Czech lamguage) on the life and work of Rychl k; see also her paper 2000] written in English. 27 I do not know who coined the word \prolongation" in this context; one could also say \extension" of a valuation to an algebraic extension eld. By the way, Rychl k himself does not use any of these words; instead he says: \We obtain a valuation of every algebraic extension of the complete eld K ". (Wir erhalten eine Bewertung jeder algebraischen Erweiterung des perfekten Korpers K .) 28 We shall see below in section 5.1 that Ostrowski had veri ed this even earlier, before 1917. But he did not publish this and therefore Rychl k could not have knowledge of Ostrowski's manuscript. History of Valuation Theory, Part I 15 We work in a non-archimedean complete valued eld K . Let O be its valuation ring. Consider a poynomial f (x) 2 O x]. Suppose that f (x) splits \approximately" into two factors, which means that there exist relatively prime polynomials g0 (x); h0(x) 2 O x] of positive degrees and a small " > 0 such that kf (x) ? g0 (x)h0(x)k " : This condition is to be understood coe cient-wise, i.e., each coe cient of the polynomial on the left hand side is of value ". It is also assumed that the degree of f (x) equals the sum of the degrees of g0 (x) and h0 (x) and, morever, that the highest coe cient of f (x) is the product of the highest coe cients of g0 (x) and h0 (x). 29 If " is su ciently small then it is claimed that f (x) splits over O. For, if R = R(g0 ; h0 ) 2 O denotes the resultant of g0 and h0 we have: Hensel-Rychl k Lemma: If " < kRk2 then f (x) splits over K . More precisely, there are polynomials g(x); h(x) 2 O x] of the same degrees as g0 (x) and h0 (x) respectively, such that f (x) = g(x)h(x) ; and that kg(x) ? g0 (x)k "kRk?1 < kRk, and similarly for h(x). The latter relations say that the factors g(x) and h(x) admit g0 (x); h0 (x) as approximations, arbitrarily close if " is su ciently small. In particular it follows that f (x), under the hypothesis of this Lemma, is reducible. From this it is easy and standard to deduce the Lemma of section 2.1, page 7 for irreducible polynomials, which Kurschak had used to extend the valuation to any algebraic extension of K . In the special case when g0 (x) = x ? a0 is linear, the resultant of x ? a0 and of ) ? f( h0 (x) = f (xx ? a0 a0 ) is computed to be f 0 (a0 ) where f 0 (x) denotes the derivative of f (x). One obtains from the above Lemma: If there is a0 2 O such that kf (a0 )k " < kf 0 (a0 )k2 then there exists a root a 2 O of f (x) with ka ? a0 k "kf 0 (a0 )k?1 < kf 0 (a0 )k. Sometime this special case is called \Hensel-Rychl k Lemma" { but in fact it is mentioned explicitly in the Czech version 1919] of Rychl k's paper only, not in the German version 1923f]. In the still more special case when kf 0 (a0 )k = 1 we obtain what most of the time is now called \Hensel's Lemma": If f (x) has a simple root a in the residue eld K = O=M modulo the maximal ideal M, then a can be lifted to a simple root a 2 O of f (x). At this point we have to warn the reader that the name \Hensel's Lemma" is not used in a unique way. Di erent authors, or even the same author in di erent publications, use the name \Hensel's Lemma" for quite di erent statements. One gets the impression that every author working in valuation theory (or elsewhere) 29 This last condition is not mentioned in Rychl k's paper { probably because it is not mentioned in Hensel's book. Perhaps Rychl k had overlooked that Hensel, in his context, normalizes his polynomials such that the highest coe cients should be a power of the given prime element; then the condition is always satis ed. Rychl k's proof tacitly assumes that the condition is satised. Without the condition, the contention of the Lemma would have to be modi ed such that f admits a decomposition f = c gh where c is a certain unit in the valuation ring, and g; h satisfy the approximation properties as stated. { Rychl k's error in the statement of the lemma was corrected by Ostrowski 1934]. 16 Peter Roquette creates his own preferred version of Hensel's Lemma. This re ects the fact that the validity of Hensel's Lemma implies very strong and important structural properties which one would like to understand from various viewpoints. Most versions of Hensel's Lemma are equivalent. A good overview of various Hensel's Lemmas (including Krasner's Lemma which is due to Ostrowski) is given in Ribenboim's instructive article 1985]. But the reader should be aware of the fact that in the literature there can be found numerous other variants of Hensel's Lemma, e.g., lifting of idempotents in algebras, or lifting of simple points on varieties, etc. It would be desirable to investigate the historical development of the ideas underlying the various versions of Hensel's Lemma, not only in valuation theory but in other branches of mathematics as well. 30 The name of Kurt Hensel will be remembered in mathematics for a long time through the name of \Hensel's Lemma", whichever form it will obtain in future developments. And, we may add, this is entirely appropriate in view of the important mathematical notion of p-adic numbers with which Hensel enriched the mathematical universe. With the paper 1923f] Karel Rychl k exits the scene of general valuation theory. 31 We have not found any later publication by him on valuations, although he was still active for quite a while on the foundation of divisor theory in algebraic number elds. He continued to play a role in the development and reorganisation of university teaching in his country. We read in 2000]: Rychl k was the rst to introduce methods and concepts of \modern" abstract algebra into this country { by means of the published treatises as well as his university lectures. Rychl k was a professor at Czech Technical University but he delivered lectures also at Charles University in Prague, as a private associate professor. See 2000] where one can nd more detailed biographical information about Rychlik. 3 Valuations in Number Theory The papers by Ostrowski and Rychl k discussed in section 2 can be viewed as appendices to Kurschak's opening paper, clearing up certain points which were left open by Kurschak including the role of archimedean valuations. After that there begins the period of expansion and application. This means that on the one hand, valuation theory was systematically expanded to study the structure of valued elds in more detail. On the other hand, it was more and more realized that valuation theory can be pro tably used in various applications. Naturally, these two strings of development { expansion and application { cannot be sharply separated. Sometimes the need for a certain application led to expand the general structure theory of valuations; on the other hand, after knowing more about the structure of valued elds it became apparent that this knowledge could be applied pro tably. 30 The so-called Newton approximation algorithm from analysis can be viewed as a special method to prove Hensel's Lemma. Also, we nd that Hensel's Lemma had been stated and proved by Gau , as observed by Gunther Frei (in a forthcoming publication on the history of the theory of algebraic function elds). 31 There had been an earlier paper 1923d] in Crelle's Journal where he gave an example of a continuous but non-di erentiable function de ned in the p-adic eld of Hensel, but this does not concern us here. History of Valuation Theory, Part I 17 The foremost and rst applications belong to number theory. This has good historical reasons since the main motivation to create the notion of valuation came from Hensel's p-adic methods introduced in number theory. Valuation theory has not only produced new methods which could be pro tably used in number theoretical research, but it has also led to a change of viewpoint. For instance the transition from \local" to \global" became one of the central questions in number theory. In this section we shall discuss the rst steps which started this development. 3.1 Hasse: The Local-Global Principle. 3.1.1 The motivation. Consider the year 1920. In that year Helmut Hasse (1898{1971), a young student of 22, decided to leave his home university of Gottingen in order to go to Marburg and continue his studies with Hensel. The motivation for this decision was Hensel's book \Zahlentheorie " 1913b] where p-adic numbers are presented as a basis for an introductory course to Number Theory. Hasse had found this book in a second hand Gottingen bookshop. 32 In the foreword to his Collected Papers 1975] he recalls: Das Buch war mir vom ersten Augenblick an wegen seiner vollig neuartigen Methoden besonders reizvoll und eines grundlichen Studiums wert erschienen : : : Auf mich hatte es eine magische Anziehungskraft ausgeubt, und so ging ich nach dem \kleinen" Marburg. From the rst moment, this book was particularly appealing to me because of his completely new methods, and certainly it seemed to be worth of detailed study : : : I felt strongly attracted to it, and hence I went to the \small" Marburg. Thus this elementary and, we may say, unpretentious booklet had instantly prompted Hasse to dive deeper into p-adic number theory { which proved to be decisive for his further work, and for the further development of algebraic number theory at large. Hensel's book does not even mention Kurschak's abstract notion of valuation. 33 The p-adic number eld Q p is introduced in the Hensel way by means of p-adic power series. The valuation (additively written) of Q p appears as \order" (Ordnungszahl ) which is assigned to each p-adic number. Limits are introduced without explicit mention of the valuation, and completeness of Q p is proved directly. Hensel's Lemma does not appear since the book does not discuss nite extensions of Q p . It may not be without interest to know that, in Hensel's book, the style of writing was not Hensel's but for the most part Fraenkel's. See Fraenkel's report 1967] where he says: Es el Hensel nicht leicht, seine originellen Gedanken in einer fur den Druck geeigneten Form darzustellen, und dies war gerade meine Starke. Er bat mich also, aufgrund der Vorlesung und regelma iger Gesprache uber das Thema, das Buch niederzuschreiben, mit Ausnahme des letzten Kapitels. 32 It is possible to identify the date when Hasse had purchased the book, namely March 20, 1920. This date is written by Hasse's own hand onto the title page of his copy of the book, together with his name. (Hasse's copy is now contained in my library, thanks to Martin Kneser.) 33 Compare the publication dates: Kurschak's paper had appeared in 1912, and Hensel's book in 1913. 18 Peter Roquette Hensel had di culties to put his original ideas into a form which was suitable for printing, but that was my strong point. Therefore he asked me to write down the text of the book, following the lectures and our regular conversations about the subject, except the last chapter. 34 Hasse registered at Marburg University in May 1920 and started his seventh semester (fourth academic year). Already at the end of this month Hensel suggested to him a subject for his doctoral thesis: quadratic forms over Q and over Q p . 3.1.2 The Local-Global Principle. Already in May 1921, one year after his moving to Marburg, Hasse had completed his thesis where he introduced and proved the famous Local-Global Principle for the representation of a rational number by a given quadratic form with rational coe cients. The problem is as follows. Let a 2 Q , and f (x1 ; : : : ; xn ) be a quadratic form over Q . To nd necessary and su cient conditions, in terms of p-adic numbers, such that a is representable by f (x1 ; : : : ; xn ), i.e., that there exist x1 ; : : : ; xn 2 Q such that a = f (x1 ; : : : ; xn ) : Hasse's Local-Global Principle says that a is representable by f (x1 ; : : : ; xn ) over Q if and only if it is representable over the p-adic completions Q p for all primes p (including p1 ). In other words: the statement 9 x1 ; : : : ; 9 xn : a = f (x1 ; : : : xn ) holds over Q if and only if it holds over Q p for all p. The point is that in the complete elds Q p there are explicit arithmetic criteria available for a to be representable by the quadratic form. If the number n of variables is > 4 then, for p 6= p1 every a 2 Q is representable by f (x1 ; : : : ; xn ) in Q p 35 , so that, by the Local-Global Principle, only the representability in Q p1 = R is relevant, and this can easily be checked according to whether the quadratic form q is de nite or inde nite. If the number of variables is 4 then Hasse develops explicit criteria, involving the quadratic residue symbol, for the representability in Q p . The case of binary quadratic forms (n = 2) had been discussed in chapter XII of Hensel's book 1913b]. 36 It seems that Hensel had not been satis ed with the restriction to n = 2, and therefore he had given to his student Hasse the task to generalize this at least for 3 or 4 variables. Finally Hasse was able to deal with an arbitrary number of variables. Hasse reports in the preface to his Collected Works 1975] that the Local-Global Principle had been suggested to him by Hensel. He mentions a postcard which Hensel had sent to him, and which he (Hasse) preserves as a valuable keepsake: 37 There Hensel wrote: 34 Fraenkel's help in editing the text is duly acknowledged in the foreword of Hensel's book. By the way, also the name of Ostrowski is mentioned; he had produced the index of Hensel's book. 35 This result has led to Artin's conjecture for forms of arbitrary degree over Q p , not only quadratic forms: Every such form of degree d with n > d2 variables should admit a non-trivial zero in Q p . It seems that Artin himself never pronounced this conjecture in writing (see the preface to Artin's Collected Papers, written by S. Lang and J. Tate in 1965). Nevertheless it attracted great attention among number theorists. After a number of partial results, the conjecture was \almost" proved by Ax and Kochen in 1965], in the sense that for given degree, Artin's conjecture holds for all but nitely many prime numbers p. Their proof was remarkable since it was the rst instance where valuation theory was combined with model theory. But soon thereafter Terjanian 1966] gave counter examples to the original conjecture of Artin. 36 This is the chapter which had been edited by Hensel himself and not by Fraenkel. 37 The postcard is dated Dec 2, 1920. It is now contained in the Hasse legacy at Gottingen. History of Valuation Theory, Part I 19 Sehr geehrter Herr Hasse! : : : Ich habe immer die Idee, da da eine bestimmte Frage zu Grunde liegt. Wenn ich von einer analytischen Funktion wei , da sie an allen Stellen rationalen Charakter hat, so ist sie rational. Wenn ich bei einer Zahl dasselbe wei , da sie fur den Bereich jeder Primzahl p und fur p1 p-adisch ist, so wei ich noch nicht, ob sie eine rationale Zahl ist. Wie ware das zu erganzen? Dear Mr. Hasse! : : : I am always harboring the idea that there is a particular question at the bottom of these things. If I know of an analytic function that it is of rational type at each point, then it is a rational function. If I know the same of a number, that it is p-adic for each prime number p and for p1 , then I do not yet know that it is rational. How would this have to be amended? Hensel's question was not precisely formulated because he did not specify what he meant by \number". If \number" means \algebraic number" then his question can be answered a rmatively if the notion of a number a \to be p-adic" is interpreted such that there exists an isomorphism of the eld Q (a) into Q p . This is a consequence of Frobenius' density results on the splitting of primes in an algebraic number eld. 38 But that must have been known to Hensel. It seems that Hensel had vaguely something else in mind, where \number" and \to be p-adic" would have to be interpreted di erently. 39 In any case, Hensel's question stimulated Hasse to nd his Local-Global Principle. He writes in 1975]: Es war die Frage am Schlu dieser Mitteilung auf der Henselschen Postkarte] die mir die Augen geo net hat : : : Aus diesem Kern wuchs mir dann rasch : : : das Lokal-Global-Prinzip fur alle Darstellungsund Aquivalenzbeziehungen bei quadratischen Formen mit rationalen und dann allgemeiner auch mit algebraischen Koe zienten. So verdanke ich die Entdeckung dieses Prinzips, wie so vieles andere, meinem verehrten Lehrer und spateren vaterlichen Freunde Kurt Hensel. It was the question at the end of this message on Hensel's postcard ] which opened my eyes : : : From this seed there grew quickly : : : the Local-Global Principle for all representation- and equivalence relations for quadratic forms with rational and also with algebraic coe cients. Thus I owe the discovery of this principle, like so many other things, to my respected teacher and later my paternal friend, Kurt Hensel. After his thesis, Hasse published in quick succession six other important papers (one jointly with Hensel) elaborating on the subject. In the rst of those papers 1923a] he developed a Local-Global Principle for the equivalence of quadratic forms over Q . This deals with the following question: Two quadratic forms f1 (x1 ; : : : xn ) and f2 (x1 ; : : : xn ) with the same number n of variables, are called \equivalent" if one can be transformed into the other by means of a non-singular linear transformation of the variables. Here, the coe cients of the quadratic forms are supposed to be contained in a given base eld, and the entries 38 Later, Frobenius' results were sharpened by Chebotarev's density theorem. We refer to x 24 of Hasse's class eld report 1930]. 39 One is reminded of the similar words which Hensel had used 1905 in Meran when he presented his \proof" of the transcendency of the number e by means of p-adic methods. See the exposition by Peter Ullrich in 1998d]. Although Hensel's \proof" turned out to be erroneous, the broad idea of it was vindicated by Bezivin and Robba 1989]. 20 Peter Roquette of the n n transformation matrix should also be in that eld. In the situation of Hasse's paper, the base eld may be either Q or the completion Q p with respect to any prime p of Q . Given two quadratic forms with coe cients in Q , Hasse proved that they are equivalent over Q if and only if they are equivalent over Q p for every prime p (including p = 1). If we denote equivalence with the symbol then this means that the statement f1 (x1 ; : : : xn ) f2 (x1 ; : : : xn ) : holds over Q if and only if it holds over Q p for all p. Here again, the point is that in Q p there are explicit arithmetic criteria available for two quadratic forms to be equivalent. This subject had been studied earlier by Minkowski in 1890]; therefore Hasse's theorem is sometimes called the \Hasse-Minkowski theorem." However, it should be kept in mind that Minkowski did not work in the p-adic completions and accordingly he did not formulate a Local-Global Principle for those. Hasse in 1923a] cites Minkowski 1890] but he does not use Minkowski's result because, he says, Minkowski relies on his systematic theory of quadratic forms over the ring of integers Z which he (Hasse) does not need and which also seems inadequate for the present purpose since the problem is of rational and not integral nature. Instead, Hasse introduces new local invariants which are su cient to characterize a quadratic form over Q p up to equivalence. Besides of the number of variables n and the discriminant d 6= 0 (modulo squares) he introduces what today is called the \Hasse invariant " of a quadratic form. If the form f is transformed into diagonal form X f (x1 ; : : : ; xn ) = ai x2 with ai 2 Q p i 1 in (which is always possible) then the Hasse invariant is given by Y ai ; a j p ij where ap;b denotes the quadratic Hilbert symbol for p which takes the value +1 p or ?1 according to whether a is a norm from Q p ( b) or not. In today's terminology we usually prefer to talk about \quadratic vector spaces" rather than quadratic forms, and then the notion of equivalence of quadratic forms corresponds to the notion of isomorphism of quadratic spaces. 40 Thus the above Local-Global Principle can also be stated in the following form: Two quadratic spaces V and W over Q are isomorphic if and only if all their localizations Vp = V Q Q p and Wp = W Q Q p are cp (f ) = isomorphic. 41 40 The idea to consider quadratic spaces instead of quadratic forms is due to Witt 1937]. In his introduction Witt refers explicitly to Hasse's papers on quadratic forms. 41 By the way, Hasse's rst Local-Global Principle in 1923] can also be formulated in terms of quadratic spaces. For, the representability of a number a by a quadratic form f (x1 ; : : : ; xn ) is equivalent with the non-trivial representability of 0 by the quadratic form f 0 (x1 ; : : : ; xn+1 ) = f (x1 ; : : : xn ) ? ax2 +1 , which means that the quadratic space V 0 belonging to f 0 is isotropic . n Hence the Local-Global Principle of 1923] can be expressed as follows: A quadratic space over Q is isotropic if and only if all its localizations are isotropic. History of Valuation Theory, Part I 21 The Hasse invariants cp (f ) appear now as invariants of the localized quadratic vector spaces V and therefore should be denoted by cp (V ). In his next paper 1924a] Hasse deals with a Local-Global Principle for symmetric matrices over Q . Thereafter he takes the important and non-trivial step to generalize all these Local-Global Principles from the base eld Q to an arbitrary algebraic number eld K of nite degree and its primes p . This is done in 1924b], 1924c], after having dealt with preparatory theorems on the quadratic residue symbol in number elds in 1923b] and 1923c]. 42 We want to emphasize that the above mentioned Hasse papers are of a di erent type from those of Kurschak, Ostrowski and Rychl k which we discussed in the foregoing sections. The latter are dealing with fundamentals, intending to provide a solid foundation for general valuation theory. But Hasse's rst papers do not care about foundations. Studying those papers we nd that Hasse does not even mention there the abstract notion of valuation. Of course, he investigates the completions Kp of a number eld K with respect to its primes p de ned by valuations (including the in nite primes p1 ). But he takes the complete elds Kp for granted, regardless of whether they are constructed via p-adic power series (as did Hensel) or by Cantor's method (as did Kurschak). In other words: Hasse applied valuation theory, whereas Kurschak, Ostrowski and Rychl k had started to build the proper foundation for it. In view of these papers of Hasse, the theory of Hensel's p-adic elds (i.e., the completions of algebraic number elds at the various primes de ned by valuations) became rmly and successfully established as a useful and powerful tool in number theory. Today this is not questioned; local elds and their structure play a dominant role. But before Hasse, this was not so obvious; there were still prominent mathematicians who considered valuation theory as not too interesting or at least super uous for number theory. Hasse reports in 1975] that Richard Courant in 1920 had voiced the opinion that Hensel's book on p-adic numbers represented a fruitless side track only (\ein unfruchtbarer Seitenweg "). It may safely be assumed that this was not only Courant's private opinion but was shared by a number of other people in his mathematical neighborhood, in Gottingen and elsewhere. 43 Let us brie y mention that Hasse's Local-Global Principle has turned out to be of importance far beyond the application to quadratic forms over number elds. For once, the Local-Global Principle was proved to hold also in various other situations in number theory, e.g., in the structure theory of central simple algebras (see section 3.4.2), and in the investigation of the embedding problem. Also in more general situations, for elds other than number elds, the Local-Global Principle has proved to be a powerful tool. Consider any multi-valued eld, i.e., a eld K equipped with a set V of valuations. Given any eld theoretic statement A over K , one can ask whether the following is true: A holds over K if and only if A holds over the completion Kp for all primes p 2 V . 42 We refer the reader to the self-contained treatment by O'Meara 1963]. There Hasse's theory of quadratic forms over number elds (and more) is presented ab ovo along Hasse's lines, preceded by an introduction to valuation theory a la Kurschak. 43 In later years Courant seems to have had revised this opinion, obviously impressed by Hasse's work not only on quadratic forms but also on other problems in number theory where valuation theory was applied. We conclude this from a letter of F.K. Schmidt to Hasse dated Oct 3, 1934 in which F.K. Schmidt reports on a conversation of Courant with Abraham Flexner, the founder of the Institute of Advanced Study in Princeton. 22 Peter Roquette If this is true then we say that A admits the Local-Global Principle over the multivalued eld K . Such a Local-Global Principle has been established, meanwhile, in quite a number of cases also outside of number theory, e.g., in the so-called eld arithmetic, and in the study of algebraic function elds over number elds. And even if, for a particular statement, the Local-Global Principle is not valid, the investigation of the obstruction to its validity often yields valuable information. It is not possible in this article to follow up the development of the valuation theoretic Local-Global Principle in full detail. This will be the topic of a separate publication, and it is an exciting story. Here, let it su ce to observe that all this started in the years 1921-22, when Hasse proved the Local-Global Principles for quadratic forms, rst over Q and then over an arbitrary algebraic number eld of nite degree. And we should not forget Hensel whose suggestion, although in rather vague terms, led Hasse to the formulation of his Local-Global Principles. 3.2 Multiplicative structure. 3.2.1 The cradle of local class eld theory. In order to obtain his Local-Global Principles for quadratic forms over a number eld K , Hasse had to study and use Hilbert's quadratic reciprocity law in K . Following up an idea of Hensel in the ;b case K = Q 1913b], he interpreted the quadratic Hilbert norm symbol ap in an arbitrary number eld K in terms of the completion Kp { where p denotes a prime of K belonging to an archimedean or non-archimedean valuation, and a; b 2 Kp . ;b As said above already, the local de nition is to put ap = 1 or ?1 according to p whether a is a norm from Kp ( b) or not. In the older literature the terminology was \norm residue symbol" instead of ;b \norm symbol", and ap was de ned for p-integers a; b 2 K only. And the ;b de nition was that ap = 1 if for any power pr , with arbitrarily large exponent, p a is congruent modulo pr to a norm from K ( b). It had been Hensel who observed that such congruence conditions for in nitely many powers of p can be viewed, in the limit, as one single condition in the completion Kp . Quite generally, observations of this kind have brought about an enormous conceptual simpli cation whose consequences, up to the present day, cannot be overestimated. The investigation of congruence properties for varying prime power modules pr can be replaced by the investigation of valuation theoretic properties of the completion Kp . Hasse con rmed Hensel's idea that the locally de ned symbol is quite suited to develop a criterion for the local representability of a number by a quadratic form. And he discovered that Hilbert's product formula Y a; b = 1 p (a; b 2 K ) (7) p provides a tool to make the transition from local to global { in conjunction with the well known ordinary product formula for the valuations themselves: Y kakp = 1 (a 2 K ) : (8) p 23 History of Valuation Theory, Part I Here, p ranges over all primes of K and k kp is the corresponding valuation in suitable normalization. ;b One of the essential properties of the Hilbert symbol ap which had to be veri ed is the \Vertauschungssatz " a; b = b; a ?1 : 44 p p (9) If p does not divide 2 (including the case when p is archimedean) then this is well known and straightforward. The only true di culty arises in the case when pj2, p i.e., when p may be wildly rami ed in Kp ( b). In the older, classical literature one tried to avoid this case by using Hilbert's product formula (7) in order to shift the ;b problem to the tamely rami ed primes. But then the explicit evaluation of ap was not possible in general, only for those a; b which satisfy certain restrictions. This was insu cient for Hasse's purpose. Finally in 1923c] he was able to show in purely local terms that a; b = (?1)L(a;b) p (10) where L(a; b) is a bilinear, non-degenerate form on the Z=2-vector space Kp =Kp 2 of square classes in Kp . And, what is essential, his construction showed that L(a; b) is symmetric which yields the Vertauschungssatz (9). This was of course well known in the case K = Q , p = 2 where the square classes of Q 2 are represented by the numbers 2x0 (?1)x1 5x2 with x = (x0 ; x1 ; x2 ) 2 (Z=2)3 . In this case L(a; b) = x0 y2 + x1 y1 + x2 y0 if the vectors x; y represent a and b respectively. But in the case of an arbitrary number eld K and an arbitrary prime p dividing 2, this was not so easy for young Hasse. Today we would deduce (10) immediately from local class eld theory. But we have to take into consideration that in 1921, local class eld theory did not yet exist. In fact, parallel to his work on the quadratic norm symbol, Hasse became interested, again on the suggestion of Hensel, in the similar problem for the m-th power norm symbol for an arbitrary positive integer m. His rst paper 1923b] on this is written jointly with Hensel. After many years of work and quite a number of papers on this topic, Hasse's results led him nally to discover local class eld theory in 1930a]. Here we cannot go into details and follow up the string of development wich nally led to the establishment of local class eld theory and its connection to global class eld theory. But we would like to state that all this started in 1923 with Hasse's papers on the norm symbols in local elds. At that time Hasse was Privatdozent at the University of Kiel. Thus Kiel in the year 1923 became the cradle of local class eld theory. 3.2.2 Hensel: The basis theorem. The problem of whether an element a 2 Kp is a norm from a given nite extension is of multiplicative nature. In order to deal with this problem it was necessary for Hasse to use the fundamental basis theorem for the multiplicative group Kp which Kurt Hensel had given already in 1916]. The general problem which Hensel dealt with can be described as follows. Consider rst the well known situation in the eld R of real numbers. In this case, the multiplicative group R admits ?1 and e as basis elements in the sense 24 Peter Roquette that every a 2 R can be written uniquely in the form a = (?1)k e where k 2 Z=2 jaj) 2 R = log( Here, e is de ned, as usual, by the exponential series 2 (11) 3 (12) exp( ) = 1 + + 2! + 3! + and e = exp(1). Formula (11) gives the structure theorem R Z=2 R (13) where on the right hand side R means the additive group of real numbers. The important feature of this isomorphism is that the multiplication in R is converted to addition in the groups Z=2 and R ; this is essentially given by the exponential function exp : R ! R >0 and its inverse, the logarithm. Now, Hensel asked whether a similar result holds in the case of a local number eld Kp with respect to a non-archimedean prime p. Is the multiplicative group Kp naturally isomorphic to some additive group which is canonically determined by the eld? He had treated this question for the rational p-adic elds Q p already in his number theory book 1913b]. One year later in 1914], Hensel started to deal with the case of an arbitrary local eld Kp which is a nite extension of Q p . The main di culty in the non-archimedean case is that the exponential series 1 (12) does not converge everywhere, it is convergent for vp ( ) > p?1 only. 45 This condition de nes an ideal of the valuation ring of Kp , say A. On A the exponential function satis es the functional equation exp( + ) = exp( ) exp( ) and de nes an isomorphism exp : A 1 + A whose inverse is given by the p-adic logarithm which is de ned through the power series expansion 3 2 (14) log( ) = ( ? 1) ? ( ? 1) + ( ? 1) 2 3 The logarithmic series converges not only for 2 1 + A but also for 2 1 + M, where M denotes the maximal ideal of the valuation ring. U1 := 1 + M is the multiplicative group of all those elements 2 Kp which satisfy 1 mod p, called the 1-units of Kp . On U1 we have log( ) = log( ) + log( ) : However, on U1 the logarithm function is in general not injective . For, there may be p-power roots of unity in Kp . It is injective on 1 + A where, as said above already, log is the inverse function of exp. Hensel in his paper 1914] considers rst the case when the rami cation degree e < p ? 1. In this case A = M and hence 1 + A = U1 . Hence every element in U1 is of the form exp( ) with 2 M. But the structure of Kp modulo U1 is well known and easy to establish: a basis is given by the elements and ! where is a prime 45 In this context vp denotes the additively written valuation of Kp , normalized in such a way that vp (p) = 1. In this normalization a prime element 2 Kp has the value vp ( ) = 1 where e e is the rami cation degree of Kp over Q p . 25 History of Valuation Theory, Part I element of Kp and ! a primitive w-th root of unity where w is the number of roots of unity in Kp . 46 Hensel concludes that every a 2 Kp has a unique representation of the form 47 8 < n2Z a = n !k exp( ) where : k 2 Z=w 2M The valuation ring of Kp is a free Zp -module of rank r = Kp : Q p ] and hence every of its non-zero ideals is so too. In particular it follows M Zr . p Using the continuous isomorphism exp : M ! U1 we see that U1 Zr too (as p Z p -modules). Hence there exist r basis elements 1 ; : : : ; r 2 U1 such that every a 2 Kp admits a unique representation of the form 8 < n2Z (15) a = n !k 1 1 r r where : k 2 Z=w This gives i 2 Zp Kp Z Z=w Zr : (16) p These formulas are the p-adic analogues to (11) and (13) in the real case. In 1914] Hensel could prove them in the case e < p ? 1 only. Hensel deals with the general case in the second part of 1914] and in 1915]. In modern terms, his argument can be summarized as follows: Consider the p-adic logarithm function log : U1 ! M. The kernel consists of the p-power roots of unity of Kp , say, ps for some integer s. The image I is a Z p -module containing A and, hence, free of rank r . Therefore there exists a section I ! U1 of the logarithm function, and this de nes an extension of the exponential function exp to I as a domain of de nition. This extension is not canonical; anyhow it follows that U1 = ps exp(I ) Z=ps Zr . The total number of roots of unity p in Kp is w = (q ? 1)ps . In this way Hensel had proved that the formulas (15) and (16) are valid in general, for an arbitrary local non-archimedean number eld Kp . But this proof does not give an e ective method to produce the basis elements i of (15). One has to choose a Zp -basis 1 ; : : : ; r of I and put i = exp( i ). This is not constructive since the module I , the image of the logarithm function, is in general not known explicitly. Moreover, the analytically de ned exponential function produces the i as possibly transcendental numbers whose arithmetic properties are not easily established. Therefore Hensel looked for a better construction of the basis elements. He found this in still another paper 1916]. There he presented an e ective construction of the basis elements i which is adapted to the natural ltration U1 U2 U3 , where U consists of those 2 Kp for which 1 mod p . This construction was used later in the work of Hasse and others to study the local norms in the context of class eld theory. 46 If e < p ? 1, the eld Kp does not contain proper p-power roots of unity and therefore w = q ? 1, the number of non-zero elements in the residue eld of Kp . 47 Hensel writes e instead of exp( ) but this may be misleading since exp(1) is not de ned and hence e is not the exponentiation by of the number exp(1). 26 Peter Roquette With this explicit construction of 1 ; : : : ; r , the basis theorem (15) is to be regarded as one of Hensel's important contributions to number theory. The ensuing structure theorem (16) deserves its place in general valuation theory, independent of its application to number theory. Note that the local number elds Kp can be abstractly characterized, independent from their origin as completions of number elds of nite degree. Namely, these elds are the locally compact valued elds of characteristic 0. 48 Sometimes the opinion is voiced that Kurt Hensel did not prove deep theorems on local elds, and that his main contribution to Number Theory was his idea that p-adic numbers exist and should be studied; the main credit for putting his ideas to work should go to Hasse and other students of Hensel. Well, Hensel's proof of the basis theorem is evidence that this view is not correct. But certainly it was Hasse who used the basis theorem in its full power in the discussion of the local Hilbert symbol, in particular in the wildly rami ed case, in connection with explicit reciprocity and class eld theory. Remark: Perhaps we should point out that the basis theorem, with a nite basis, does not hold in the analogous case of power series elds with nite coe cient elds. This seems to be one of the reasons why in the latter case, the theory of those elds has not yet been proven to be decidable, whereas in the case of local number elds this question is settled by the work of Ax-Kochen and Ershov. 49 3.3 Remarks on p-adic analysis. In the foregoing section we had occasion to mention functions which were analytic in the p-adic sense: the exponential function and the logarithm. There are other p-adically analytic functions which were successfully employed in number theory. There have been a number of attempts to develop a systematic theory of analytic functions in the context of non-archimedean valuation theory, analogous to the theory of complex valued analytic functions. This development, including Tate's theory of rigid analysis, is an indispensable part of the history of valuation theory. However, for the reasons stated in the introduction it was not possible to include it into this manuscript. Let it su ce to say that the rst paper in this direction was the thesis of Schobe 1930d], a doctorand of Hasse in Halle. For more information we refer the reader to the treatment by Ullrich 1995] and the literature cited there. In addition, the reader might consult 1985b] concerning the contributions of Marc Krasner. Also, we would like to point out that by means of p-adic analysis several proofs of transcendency have been given. The rst was by Kurt Mahler in 1932i] when he proved that the value of the p-adic exponential function exp( ) is transcendental for any algebraic in the domain of convergence of the p-adic exponential series (12). Later in 1935a] he proved the p-adic analogue of Gelfand's result: the quotient of ) p-adic logarithms log( ) , if irrational, is transcendental for algebraic p-adic numbers log( , which are in the domain of convergence of the logarithmic series (14). There followed a long series of p-adic transcendental theorems whose survey indeed would require a separate article. Hensel's idea to prove the transcendency of the real number e by p-adic analytic methods 1905a], which contained an error, has been vindicated to some extent by 48 See e.g., Pontryagin's book 1957], or Warner 1989b]. 49 See e.g. our Lecture Notes 1984]. History of Valuation Theory, Part I 27 Bezivin and Robba 1989]. For a historic review we refer to Ullrich's paper 1998d] and the literature cited there. 3.4 Valuations on skew elds. During the late 1920's and early 1930's there was growing awareness that the theory of non-commutative algebras could be used to obtain essential information about the arithmetic structure of commutative number elds. This view was forcibly and repeatedly brought forward by Emmy Noether; see e.g., her report at the Zurich International Congress of Mathematicians 1932] and the literature cited there. The story started with the appearance of Dickson's book 1923g] \Algebras and their Arithmetics " in which the author did the rst steps towards an arithmetic theory of maximal orders in an algebra over number elds. 50 This book gained increasing interest among the German algebraists 51 , in particular since Speiser had arranged a German translation 1927c]. Emil Artin presented a complete and exhaustive theory of the arithmetic of maximal orders of semisimple algebras over number elds in his papers 1928a], 1928b], 1928c]. 52 He cited Dickson as a source of inspiration but his theory went far beyond, describing such a maximal order and its groupoid of ideals (in the sense of Brandt), in a similar way as Noether had done in the commutative case for what are now called Dedekind rings. 53 Shortly afterwards Hasse 1931] gave a new treatment, and this time on a valuation theoretic basis . In the same way as a Dedekind ring can be treated by means of its localizations with respect to the valuations belonging to its prime ideals, Hasse showed that in the non-commutative case, the arithmetic of a maximal order can be similarly described by its localizations with respect to the primes of the center. Moreover, by including the in nite primes belonging to the archimedean valuations, Hasse was able to proceed much further towards non-commutative foundation of commutative number theory { following the desideratum of Emmy Noether. Thus once more Hasse showed that valuation theory provides for useful and adequate methods to deal with questions of higher algebraic number theory. This was well acknowledged by his colleagues. 54 As an example let us cite Emmy Noether, in a postcard to Hasse of June 25, 1930: Ihre hyperkomplexe p-adik hat mir sehr viel Freude gemacht : : : Your hypercomplex p-adics has given me much pleasure : : : And Artin in a letter of Nov 27, 1930: 50 The book also contained the rst text book treatment of the Wedderburn structure theorems for semisimple algebras. 51 By this I mean e.g., Emil Artin, Helmut Hasse, Emmy Noether, Andreas Speiser, Bartel van der Waerden and the people around them. Perhaps it is not super uous to state that the word \German" in this connection is not meant to have implications in the direction of political doctrines. (Artin was Austrian, Speiser was Swiss, van der Waerden was Dutch.) 52 Artin had divided his article into three separate papers, appearing successively in the same volume 5 of Hamburger Abhandlungen. 53 However Artin did not work on an axiomatic basis like Noether for Dedekind rings (she called them \Funf-Axiome-Ringe " ( ve axioms rings)). In later years one of the Ph.D. students of Artin, Karl Henke, presented an axiomatic description 1935] modeled after Noether's axiomatic treatment of Dedekind rings. 54 As mentioned in the introduction already, van der Waerden 1975a] speaks of Hasse as \Hensel's best and great propagandist of p-adic methods". Hasse, he said, came often to Gottingen and so van der Waerden was inspired by Hasse, as well as by Ostrowski, while dealing with valuations during the write up of the rst volume of his book \Moderne Algebra ". 28 Peter Roquette : : : Ich danke Ihnen auch fur die Ubersendung der Korrekturen Ihrer Arbeit uber hyperkomplexe Arithmetik. Dadurch ist wirklich alles sehr einfach geworden : : : : : : Also, I would like to thank you for sending the proof sheets of your paper on hypercomplex arithmetics. With this, everything really has become very simple : : : Let us describe Hasse's theorems, at least for local elds, in some more detail. 3.4.1 Hasse's theorems. Let K be an algebraic number eld and A a central simple algebra over K . For any prime p of K let Kp be the corresponding completion and Ap = A K Kp the localization of A. This is a central simple algebra over Kp and so, by Wedderburn's theorem, Ap is a full matrix algebra over a central division algebra Dp over Kp . Hence, as a rst step Hasse considers valuations of division algebras Dp over local elds. Suppose that p is non-archimedean. The following theorems are proved in 1931]: 1. The canonical valuation of Kp admits a unique extension to a valuation of Dp . Here, a valuation of the skew eld Dp is de ned by Kurschak's axioms (1){(4) in the same way as for commutative elds. The formula (5) for the extensions is also valid in the non-commutative case; but usually one replaces the norm by the socalled reduced norm for Dp jKp . The proof of 1. is quite the same as in the case of a commutative extension eld, using Hensel's Lemma. The rami cation degree e and the residue degree f of Dp jKp are de ned as in the commutative case. Then: 2. The rami cation degree e and the residue degree f of Dp jKp are both equal to the index m of Dp , so that Dp : Kp ] = ef = m2 . The surprising thing is that, on the one hand, statement 1. is valid for division algebras in the same way as for commutative eld extensions, whereas statement 2. shows a completely di erent behavior of valuations of division algebras when compared to commutative eld extensions. Namely, rami cation degree and residue eld degree are both determined by the degree m2 of Dp . This has important consequences. Let be a prime element and ! a primitive q ? 1-th root of Kp (where q is the number of elements in the residue eld). Let m (X ) denote an irreducible polynomial of degree m which divides X qm ?1 ? 1 (such divisor does exist since there is an unrami ed extension of Kp of degree m). Then: 3. Dp = Kp (u; ) is generated by two elements u and with the de ning relations: um = ; u?1 u = qr where r is some integer prime to m, uniquely determined by Dp modulo m. In particular it is seen that Dp is a cyclic algebra. Moreover, Dp is uniquely r determined by the index m and the number r modulo m. The quotient jp (Dp ) := m modulo 1 is called the Hasse invariant of the skew eld Dp . This theorem then leads to the determination of the Brauer group Br(Kp ) of all central division algebras over Kp or, equivalently, of the similarity classes of m( ) = 0 ; History of Valuation Theory, Part I 29 central simple algebras over Kp . Using the above theorems, Hasse could prove that Br(Kp) Q =Z (17) and this isomorphism is obtained by assigning to each division algebra Dp jKp its Hasse invariant. If Kp is a local eld for an in nite (archimedean) prime then Kp = R or Kp = C . In the rst case there is only one non-trivial skew eld with center R , 1 namely the quaternion eld H . To this is assigned the Hasse invariant jp (H ) = 2 , 1 Z =Z is of order 2. For the complex eld we have Br (K ) = 0. and so Br(R ) 2 p Remark. Hasse's paper 1931] was the rst one where valuations of division algebras have been systematically constructed and investigated, on the basis of Kurschak's axioms. This started a long series of investigations of valuations of arbitrary division algebras over a valued eld, not necessarily over a local number eld. We refer to the excellent report by A. W. Wadsworth 2002]. 3.4.2 Consequences. Hasse's theorems have important implications for local class eld theory. Observe that just one year earlier the main theorems on local class eld theory had been presented by Hasse 1930a] and F. K. Schmidt 1930b] as a consequence of global class eld theory. 55 But from the very rst discovery of local class eld theory, Hasse had looked for a foundation of local class eld theory on purely local terms, independent from global class eld theory. Emmy Noether shared this opinion; let us cite from the postcard (mentioned above already) of Noether to Hasse of June 25, 1930, when Hasse had sent her the manuscript of 1931]: : : : Aus der Klassenkorpertheorie im Kleinen folgt: Ist Z zyklisch n-ten Grades uber einem p-adischen Grundkorper K , so gibt es in K mindestens ein Element a 6= 0, derart da erst an Norm eines Z Elementes wird. Konnen Sie das direkt beweisen ? Dann konnte man aus Ihren Schiefkorperergebnissen umgekehrt Klassenkorpertheorie im Kleinen begrunden : : : From local class eld theory follows: If Z is cyclic of degree n over a p-adic base eld K then there exists at least one element a 6= 0 in K such that only the an is a norm of a Z -element. Can you prove this directly ? If so then one could deduce from your skew eld results backwards the local class eld theory : : : In fact, it is easy and straightforward to answer Noether's question positively, using Hasse's result 3. on skew elds. On the other hand, it seems that Emmy Noether had jumped too early to the conclusion that this alone already provides a foundation for the whole body of local class eld theory. The inclusion of arbitrary abelian eld extensions instead of cyclic ones required some more work. Later, Noether admitted this in her Zurich address 1932]. After referring to Hasse's canonical de nition of his norm symbol (which he could manage because of the theorems above) she mentions : : : eine hyperkomplexe Begrundung der Klassenkorpertheorie im Kleinen, auf derselben Grundlage beruhend, die neuerdings Chevalley gegeben hat, wobei aber noch neue algebraische Satze uber Faktorensysteme zu entwickeln waren. 55 Hasse's 1930a] had been received by the editors on March 16, 1929 while 1931] was received on June 18, 1930. 30 Peter Roquette : : : a hypercomplex foundation of local class eld theory, based on the same principles, which has recently been given by Chevalley, where however, in addition, new algebraic theorems on factor systems had to be developed. The \new algebraic theorems on factor sets" can be found in Chevalley's Crelle paper 1933e]. From today's viewpoint, these \theorems" belong to the standard prerequisites of Galois cohomology but in those times cohomology theory had not yet been established on an abstract level. Based on his local results, Hasse was able in a subsequent paper 1932e] to determine the Brauer group Br(K ) also of a global number eld K , by means of valuation theoretic notions. Let p range over all primes of K and I denote the direct sum of all local groups of Hasse invariants, i.e., X X1 I= Q =Z Z =Z : p nite p real 2 Then we have a natural map j : Br(K ) ! I, assigning to each central simple algebra A over K the vector of the Hasse invariants of its local components. More precisely, let Ap = A K Kp , and let Dp be the division algebra which is similar to Ap , so that Ap is a full matrix algebra over Dp . Then jp (A) is de ned to be the Hasse invariant jp (Dp ). And j (A) 2 I is the vector consisting of the components jp (A). In addition, there is the natural map s : I ! Q =Z by adding the components of every vector of I. Then the main theorems of 1932e] can be expressed by saying that the following sequence is exact: j s 0 ! Br(K ) ?! I ?! Q =Z ! 0 (18) The exactness at the term Br(K ) means that the map j is injective. This fact is the famous Local-Global Principle for central simple algebras. It had been rst proved in a joint paper of Hasse with Richard Brauer and Emmy Noether 1932c], and it was based on class eld theory. 56 The exactness at the term I says that the local invariants of a central simple algebra A over K satisfy the relation X jp (A) 0 mod 1 (19) p and that this is the only relation between the local Hasse invariants. Many years later, Artin and Tate presented in their Seminar Notes 1952a] an axiomatic foundation of class eld theory. Their axioms were given in the language of cohomology which by then was well developed. There are two main axioms. Their Axiom I is essentially the cohomological version of the exactness in (18) at the term Br(K ). And their Axiom II is essentially coming from the exactness at I. Thus the work of Hasse, Brauer and Noether which led to the exact sequence (18), had become the very base for the development of class eld theory in an axiomatic framework. And this had been achieved by following the ideas of Kurschak and Hensel about valuations and their localizations. 56 Another proof was given later by Zorn 1933d], based on analytic number theory as developed in the thesis of Kathe Hey 1929]; both had been Ph.D. students of Artin in Hamburg. History of Valuation Theory, Part I 31 3.5 Artin-Whaples: Axioms for Global Number Theory. As we have seen, the introduction of valuation theory into number theory has brought about drastic changes of view point and enormous advances. There arose the question whether this could be explained somehow. What are the special valuation theoretic properties which characterize number elds in contrast to other elds ? If K is a eld and p a prime of K then we denote by k kp a real valued valuation belonging to p. Suppose that K has the following properties: I. For any a 2 K , there are only nitely many primes p of K such that kakp 6= 1. II. Every prime of K is either archimedean or discrete with nite residue eld. 57 Then, if K is of characteristic 0, it follows that K is an algebraic number eld of nite degree. If K is of characteristic p > 0 then K is a nitely generated algebraic function eld of one variable over a nite eld { if it is assumed that K admits at least one prime to avoid trivialities. Today these elds are called \global elds ". Thus global elds can be characterized by the valuation theoretic properties I. and and II. The proof is quite easy. The above characterization of global elds was included in Hasse's textbook \Zahlentheorie " 1949]. That book had been completed in 1938 already but due to external circumstances it could appear in 1949 only. Thus its contents represent more or less the state of the art in 1938, at least in its rst edition. In the second edition (which appeared in 1963) Hasse added a proof of the theorem of ArtinWhaples which yields a stronger and much more striking result. Emil Artin had announced this theorem in an address delivered on April 23, 1943 to the Chicago meeting of the American Mathematical Society. The published version, jointly with his student George Whaples appeared in 1945]. Their main result can be regarded in some sense to be the nal justi cation of valuation theory in its application to number theory. Artin considers a eld K equipped with a non-empty set of primes S , which may or may not be the set of all primes of K . His axioms are: I:0 Axiom I. holds for primes which are contained in S , i.e., for any a 2 K there are only nitely many p 2 S such that kakp 6= 1. II:0 Axiom II. holds for at least one prime p 2 S , i.e., p is archimedean or discrete with nite residue eld. III:0 In addition, the valuations k kp for p 2 S satisfy the product formula Y kakp = 1 (a 2 K ) : (20) p2 S If these axioms are satis ed then again, K is either an algebraic number eld of nite degree, or a nitely generated algebraic function eld of one variable over a nite eld of constants. 57 Both cases can be subsumed under the uni ed condition that the completion Kp is locally compact with respect to the topology induced by the valuation. 32 Peter Roquette Moreover, the set S indeed consists of all primes of K , and the selected valuations k kp are uniquely determined up to a substitution of the form k kp 7! k kr p where the exponent r is independent of p. 58 In view of this theorem it is possible, in principle, to derive all theorems of number theory from the Artin-Whaples valuation theoretic axioms. In fact, the authors do this for two of the fundamental theorems of number theory, namely Dirichlet's unit theorem and the niteness of class number. The reader is struck not only by the economy but also by the beauty of those proofs. We observe that not only algebraic number elds appear in Artin-Whaples' theorem but also algebraic function elds with nite eld of constants. This re ects a long standing observation, going back to Dedekind and Gauss, on the analogy between algebraic number theory and the theory of algebraic functions with nite base eld. 59 The authors also discuss a somewhat more general setting, in which K contains a base eld k for S , which is to say that all primes p 2 S are trivial on k. In this case it is required that at least one p 2 S is discrete, and its residue eld is of nite degree over the base eld k. 60 But k is not necessarily nite. In this more general case, the product formula (20) implies that K is a eld of algebraic functions of one variable over k, and that S consists of all primes of K over k. 4 Building the foundations In the foregoing section 3 we have reported about the impact of valuation theory to number theory during the 1920's and 1930's. Because of those striking results there was growing interest to develop general valuation theory beyond the rst steps which we discussed in section 2. The motivation for this direction of research was to create an arsenal of new notions and methods which could be pro tably applied to situations other than number elds, e.g., function elds of one or more variables, elds of power series and of Dirichlet series, functional analysis, but also to number elds of in nite degree { and more. 4.1 Dedekind-Hilbert theory. Let LjK be a Galois eld extension whose base eld K carries a prime p corresponding to a non-archimedean valuation. Let us x one prime P of L which is an extension of p. The Dedekind-Hilbert theory establishes a connection of the structure of the Galois group G and the structure of P as an extension of p. This manifests itself by a decreasing sequence of subgroups G Z T V = V1 Vi Vi+1 1 whose members are de ned by valuation theoretic conditions with respect to P. Here, Z , T and V are the decomposition group (Zerlegungsgruppe ), the inertia group (Tragheitsgruppe ) and the rami cation group (Verzweigungsgruppe ) of P. These three groups are the \basic local groups" of P whereas the Vi are the \higher rami cation groups". 61 Usually one puts V0 = T and V?1 = Z . The groups T; V and the Vi are normal in Z . III:0 58 Later 1946] the authors showed that Axiom I:0 could be omitted if the product formula is understood to imply that for every a 2 K the product (20) was absolutely convergent with the limit 1. But the result remains the same. 59 See our manuscript 1998b], and also a forthcoming manuscript of Gunther Frei concerning Gauss' contributions to this question. See also Ullrich's manuscript 1999a]. 60 This is equivalent to saying that the completion Kp is locally linearly compact over k. 61 Hilbert 1894] called them the \mehrfach uberstrichene Verzweigungsgruppen." History of Valuation Theory, Part I 33 The rami cation groups re ect the decomposition type of the prime p of the base eld in the extension LjK . Hilbert 1894] had developed the theory of these rami cation groups in case K and k are number elds of nite degree. 62 Since then one often speaks of \Hilbert theory". Deuring 1931a] however says \Dedekind-Hilbert theory". Indeed, Dedekind had already arrived much earlier at the same theory, but restricted to the basic local groups Z; T; V , without the higher ram cation groups Vi . However he had published this only after Hilbert in 1894a]. 63 Here, let us follow Deuring and use \Dedekind-Hilbert theory". This seems particularly adequate in this context since in general valuation theory, the higher rami cation groups (which are due to Hilbert) do not play a dominant role, and often cannot be de ned in a satisfying way; thus our discussion will be mostly concerned with the basic local groups (which are due to Dedekind). Hilbert had included this theory in his \Zahlbericht " 1897]. Hence we may safely assume that the Dedekind-Hilbert rami cation theory was well known among the number theorists during that time and did not need any explanation or motivation. 4.1.1 Krull, Deuring. In the years 1930-1931 there appeared two papers almost simultaneously, which generalized Dedekind-Hilbert theory to an arbitrary Galois eld extension LjK and to an arbitrary non-archimedean prime p of K , including the case when p is non-discrete: Wolfgang Krull 1930c] and Max Deuring 1931a]. Each of the two authors mentions the other paper in his foreword and states that his paper was written independently. The mere fact that two authors worked on the same subject at the same time, may indicate a widespread feeling of the necessity to have Dedekind-Hilbert theory available in the framework of general valuation theory. Deuring was a student of Emmy Noether 64 and Krull too was close to the circle around her. 65 We know that Emmy Noether always freely discussed her ideas with whoever was listening to her; hence it is conceivable that both, Deuring and Krull, were stimulated by her to deal with this problem. Both authors came to similar conclusions. It turned out that the theorems in the general case are essentially the same as in the classical case considered by Dedekind and Hilbert, at least for the basic local groups Z; T; V { with some natural modi cations though. In today's terminology the following was proved. 1. All primes P0 of L which extend the given prime p of K are conjugate to P under the Galois group G. More precisely, those P0 are in 1 ? 1 correspondence with the cosets of G modulo the decomposition group Z . Consequently, the index (G : Z ) equals the number r of di erent primes of L extending p. This is precisely as in the classical case of number elds. 62 More precisely, Hilbert assumed that k = Q . But it is understood that from this, the case of an arbitrary number eld of nite degree is immediate. 63 See also the literature mentioned by Ore in Dedekind's Gesammelte mathematische Werke , vol.2 p.48. 64 Deuring's paper 1931a] was not yet his doctoral thesis. It was written before Deuring got his Ph.D. 65 Krull held a position as associate professor in Erlangen at the time when he wrote this paper. 34 Peter Roquette In the following let K and L denote the residue elds of p and of P respectively and f = L : K ] the residue degree. Let fsep denote its separable part, i.e., fsep = Lsep : K ] . Similarly fins = L : Lsep ] denotes the inseparable part of f . 2. Lsep is a Galois eld extension of K . Each 2 Z induces an automorphism of LjK , and this de nes an isomorphism of the factor group Z=T with the full Galois group of LjK . Consequently (Z : T ) = fsep . In the classical case Z=T is cyclic. This re ects the fact that in the classical case K is nite and hence admits only cyclic extensions. In the following let ? and denote the value groups of the valuations belonging to p and P, so that ? . The factor group =? is a torsion group; its order e = ( : ?) is the rami cation degree of P over p. Let e0 denote the part of e which is relatively prime to the residue characteristic p, thus e0 = ( 0 : ?) where 0 denotes the group of those elements in whose order modulo ? is prime to p . Similarly ep = ( : 0 ) is the p-part of e. 66 3. For 2 T and a 2 L let (a) be the residue class of a ?1 . This induces a character 2 Hom( 0 =?; L ), and 7! denes an isomorphism of T=V with the full character group of 0 =?. Consequently T=V is abelian and (T : V ) = e0 . In the classical case T=V is cyclic. This re ects the fact that in the classical case P is discrete which implies Z , hence every proper factor group of is cyclic. Both Krull and Deuring found it remarkable that in the general case T=V is not necessarily cyclic, and they produced examples. 4. The rami cation group V is a p-group and hence solvable. Its order jV j is a multiple of ep fins . 67 In the classical case, jV j = ep . Both Krull and Deuring state that jV j may be a proper multiple of ep fins . But they did not introduce the quotient j = e jVfj = ejZf p ins (which is a p-power) as an invariant which was worthwhile to study. It was later called by Ostrowski the defect of P over p. 68 Thus we have nally L : K] = e f r (21) where r is (as above) the number of di erent primes P0 which extend p. 69 Deuring gave also a de nition of higher rami cation groups but it seems that these, except in the discrete case where they were introduced by Hilbert, did not play a signi cant role in future developments. (But see below in section 4.1.2 about Herbrand's work.) 66 If the residue eld is of characteristic 0 then 0 = , e0 = e, ep = 1, V = 1. 67 In view of this it seems to be adequate, and is often done nowadays, to view fins as part of the rami cation degree and, accordingly, call fsep the residue degree. 68 See section 5. 69 Sometimes in the literature, the defect is denoted by d instead of (although the letter d is often reserved for the discriminant). Also, the number r may be denoted by g (although the letter g is often reserved for the genus). With this notation, if n denotes the eld degree, formula (21) acquires the cute form n = d e f g . History of Valuation Theory, Part I 35 Remark. Deuring's paper contains a new proof of the fact that a prime p of K can be extended to any algebraic extension L of K . Of course, this had been proved in the very rst paper on valuation theory by Kurschak 1913], with the formula (5) giving explicitly the value of an algebraic element over K . Deuring's proof does not use the valuation function but rather the valuation ring O belonging to the prime p. Deuring rst observed that O is a maximal subring of K . 70 Conversely, every maximal subring of K belongs to a prime of K . Now, given an algebraic extension LjK , Deuring considers a subring OL L which is maximal with the property that OL \ K = O ; 71 then he showed that OL is a maximal subring of L, hence belongs to a prime P of L extending p. In this proof we see clearly the in uence of Emmy Noether who preferred the ring-theoretic viewpoint. The extreme simplicity and elegance of this proof stands against the fact that it is not constructive { contrary to Kurschak's proof. It should be mentioned, however, that the method of Deuring's proof is the same which today is used in the existence proof for arbitrary Krull valuations which are centered at a given prime ideal of a given ring. Deuring does this existence proof for arbitrary algebraic extensions LjK including those of in nite degree. Accordingly, statement 1. also covers in nite Galois extensions. It seems to us that in the other statements too he would have liked to include in nite extensions but somehow hesitated to do so. Note that Krull's paper 1928] on the pro nite topology of in nite Galois groups had appeared not long ago, and so Deuring may have thought that some readers of his article would not yet be acquainted with compact topological groups as Galois groups. But it is surprising that Krull too considered nite extension only; after all, he had shown the way how to handle in nite Galois extensions in the rst place. Moreover, Krull did not discuss higher rami cation groups which is the only instance where the generalization of Dedekind-Hilbert theory to in nite Galois extensions is not straightforward. 4.1.2 Herbrand. Soon later Jacques Herbrand lled this gap in 1932b], 1933f]. 72 He fully realized that in number theory one would have to consider in nite extensions too. His method is quite natural, namely he considers in nite extensions as limits of nite ones. He restricts his discussion to in nite extensions of number elds. In this case the value groups of the non-archimedean valuations, although not necessarily discrete, have rational rank 1. There was one di culty, however, which is connected with the enumeration of the higher rami cation groups Vi . Consider the case where the value group ? is discrete. If LjK is of nite degree then is discrete too. Let denote a prime element for P and normalize the corresponding additive valuation such that vP ( ) = 1. Then the i-th rami cation group Vi consists by de nition of those 2 V for which vP ( ?1 ? 1) i. Now let M jK be a Galois subextension of LjK with Galois group H which is a factor group of G. It is true that the image of Vi in H is some higher rami cation group for M jK but in general it is not the i-th one. Herbrand had given in 1931b] a method how to change the enumeration of the rami cation groups such as to become coherent with the projection of G onto its 70 This depends on the fact that the value group is of rank 1. For arbitrary Krull valuations this is not true. 71 The existence of such OL would today be proved by Zorn's Lemma; at the time of Deuring's paper he used the equivalent theorem that every set can be well ordered. 72 These papers were published posthumously, with a preface of Emmy Noether. In August 1931 Jacques Herbrand had died in a fatal accident in the mountains. 36 Peter Roquette factor groups H . Today we would describe it by the Hasse-Herbrand function '(t) = Zt dx 0 (V0 : Vx ) where Vx is de ned for real x ?1 to be Vi when i is the smallest integer x. (Here, x 2 R ; x ?1.) This function '(t) is piecewise linear, monotonous and continuous; let (u) denote its inverse function. Then put Vt = V '(t) ; hence V u = V (u) : This is called the \upper enumeration" of the rami cation groups; note that u 2 R is not necessarily an integer. With this notation, Herbrand's theorem now says the following. Consider the situation as described above, i.e., H is a factor group of G, the Galois group of a Galois subextension M jK of LjK . Then: The image of V u G in the factor group H is the u-th rami cation group (in upper enumeration) of M jK . Thus Herbrand's upper enumeration of the rami cation groups is coherent with respect to projections G ! H . Hence, for an in nite Galois extension LjK Herbrand can de ne the V u as the projective limit of the corresponding rami cation groups (upper enumeration) of the nite factor groups of G. This method works for any discrete valued eld K with perfect residue eld, and any Galois extension LjK , nite or in nite. 73 Remark: As we know today, the full power of the Dedekind-Hilbert theory unfolds itself only if in nite Galois extensions are taken into consideration. For instance, we may take for L the separable algebraic closure of K . In that case G = GK is the absolute Galois group of the eld K . For an algebraic number eld K of nite degree it has been shown by Neukirch 1969] that the whole arithmetic structure of K , which manifests itself by the primes p of K and their interconnections, is already coded in the absolute Galois group GK and the structure of its basic local groups belonging to the various primes. Similar result in 1969a] for the Galois group of the maximal solvable extension. In this connection we would also like to mention the work of Florian Pop who proved similar statements for arbitrary nitely generated in nite elds, not necessarily algebraic. For a survey see 1997]. The origin of all this newer development can be seen in the papers by Krull, Deuring and Herbrand of 1931{1933. 4.2 F. K. Schmidt: Uniqueness theorem for complete elds. We have mentioned above Neukirch's result about the determination of an algebraic number eld K of nite degree by its absolute Galois group GK . There are two main ingredients (among others) in Neukirch's proof: One is Dedekind-Hilbert theory of rami cation for in nite algebraic extensions as discussed above. The other is the uniqueness theorem of F. K. Schmidt in 1933] which says: Uniqueness theorem. Let K be a complete valued eld and suppose that K is not algebraically closed. Then the valuation of K is unique in the sense that it is the only valuation (up to equivalence) in which K is complete. For any other valuation of K (not equivalent to the given one) its completion is algebraically closed. 73 I. Zhukov 1998e] has given a modi ed method which in certain cases also works if the residue eld is not perfect. History of Valuation Theory, Part I 37 The paper carries the title \Mehrfach perfekte Korper " (multi-complete elds) which means \ elds which are complete with respect to more than one valuation". This title is somewhat misleading because F. K. Schmidt's theorem says that a complete eld is never multi-complete except in the trivial case when K is algebraically closed. 74 However the title can be understood from the original aim of F.K. Schmidt's work. Let us cite from letters of F. K. Schmidt to Hasse. In early 1930 F. K. Schmidt (who held a position of Privatdozent in Erlangen) had visited Hasse (who at that time was full professor in Halle). After his return to Erlangen F. K. Schmidt wrote on Feb 14, 1930 to Hasse: Was ich Ihnen uber die Fragen, die wir in Halle erorterten, schreiben wollte, betri t zweierlei: Einmal handelt es sich um den Satz : Ist der Korper K hinsichtlich der diskreten Bewertung v perfekt, so ist v 1. die einzige diskrete Bewertung von K , 2. die einzige Bewertung von K hinsichtlich der K perfekt ist : : : Regarding the questions which we discussed in Halle, I wanted to write you two things: For one, there is the theorem: If a eld K is complete with respect to a discrete valuation v then v is 1. the only discrete valuation of K , 2. the only valuation of K with respect to which K is complete ::: Here, \only" is meant to be \only up to equivalence". From this letter it appears that F.K. Schmidt started this work on the suggestion of Hasse. In any case, this work was done parallel to his joint work with Hasse, determining the structure of complete discrete valued elds 75 , and in this connection it was of course of high interest to know whether a eld could be complete with respect to two di erent valuations. At the end of the letter, after sketching his proof of the above statements 1. and 2., F.K. Schmidt says: Ich mochte mir nun noch uberlegen, ob ein Korper hinsichtlich zweier verschiedener Bewertungen, die dann naturlich nicht diskret sind, perfekt sein kann, und falls das moglich ist, die Bescha enheit eines solchen Korpers kennzeichnen. Next I wish to investigate whether a eld could be complete with respect to two di erent valuations, which of course cannot be discrete. And if this is possible then I want to characterize the structure of such eld. We conclude that F.K. Schmidt apparently did not have a conjecture what to expect for non-discret valuations and, indeed, he was looking for a characterization of those elds which are complete with respect to two di erent valuations, expecting possibly non-trivial instances of this situation { as the title of the paper suggests. 74 An algebraically closed eld whose cardinality is su ciently large is always multi-complete. 75 See section 4.3. This topic was the second of the two things which F. K. Schmidt wanted to discuss in his letter to Hasse. 38 Peter Roquette Already one month later, on March 29, 1930, F.K. Schmidt writes that he has solved the problem; he states his theorem as given above and sends a corresponding manuscript to Hasse. 76 The main part of F. K. Schmidt's proof is based on Hensel's Lemma, hence is valid for arbitrary Henselian elds. The completeness property is used only in the proof of the following lemma: If a complete eld K is separably closed then it is algebraically closed. 77 Indeed, for a 2 K consider the separable polynomials xp ? cx ? a where p is the characteristic and the parameter c 2 K . If c converges to 0 then the roots of xp ? cx ? a converge to the p-th root of a and therefore a1=p 2 K . All the other arguments in F. K. Schmidt's proof are valid for arbitrary Henselian elds. Thus his proof in 1933] yields the following theorem: The above uniqueness theorem remains valid if \complete" is replaced by \Henselian", and \algebraically closed" by \separably closed". 78 However, F. K. Schmidt does not use the notion of Henselian eld; this notion appeared only in Ostrowski's paper 1934] under the name of \relatively complete" eld (see section 5). We have mentioned the uniqueness for Henselian elds because it was stated and proved by Kaplansky and Schilling in 1942]. They reduced it to the uniqueness theorem for complete elds, citing F. K. Schmidt's paper 1933] { apparently without noticing that F. K. Schmidt's proof itself yields the result for Henselian elds too. 79 4.2.1 The Approximation Theorem. Today, the proof of F. K. Schmidt's uniqueness theorem is usually presented with the help of the following Approximation Theorem. Let K be a eld and let p1 ; : : : ; pr be di erent primes of K , with valuations k k1 ; : : : ; k kr respectively. Given any elements a1 ; : : : ; ar 2 K there exists x 2 K which approximates each ai arbitrarily close with respect to pi . This means that kx ? ai ki < "i (1 i r) for arbitrary prescribed numbers "i 2 R . In order to deduce the uniqueness theorem from this, consider a eld K which is Henselian with respect to two di erent primes p1 and p2 . One has to prove that K is separably closed, i.e., that every separable polynomial splits completely over K . Consider two monic separable polynomials f1 (X ); f2(X ) of the same degree, say n. From the approximation theorem we conclude that there exists a monic polynomial g(X ) of degree n which approximates f1 (X ) with respect to p1 and f2 (X ) with respect to p2 , coe cientwise and arbitrarily close. Now from Hensel's 76 It took more than 2 years until F. K. Schmidt submitted the nal version for publication. From the correspondence it appears that he was frequently changing and rewriting his manuscript until nally it was in a form which satis ed his sense of elegance. 77 As mentioned in section 2.2.1 this result had been already obtained by Ostrowski in his rst paper 1913a]. F. K. Schmidt uses the same argument as did Ostrowski. 78 According to F. K. Schmidt's proof, this theorem includes the case of a eld with an archimedean valuation, if in such case the property \Henselian" is interpreted as being isomorphic to a real closed or algebraically closed sub eld of C . 79 Kaplansky and Schilling consider non-archimedean valuations only but F. K. Schmidt's theorem includes archimedean valuations too, as pointed out above already. History of Valuation Theory, Part I 39 Lemma (for p1 ) it follows that after su ciently close p1 -approximation, g(X ) has the same splitting behavior over L as does f1 (X ). Similarly for p2 , and so g(X ) has also the same splitting behavior as f2 (X ). We conclude that every two monic separable polynomials f1 (X ); f2 (X ) 2 K X ] have the same splitting behavior over K . But certainly there do exist, for any degree n, separable monic polynomials f (X ) which Q completely. One has to choose n di erent elements ai 2 K and split put f (X ) = 1 i n (X ? ai ). However, F. K. Schmidt did not use the general approximation theorem, perhaps because he was not aware of its validity. Instead, he gave an ad hoc construction for the approximating polynomial g(X ) to be able to conclude as we have done above. Remark. According to our knowledge, the rst instance where the approximation theorem had been formulated and proved, including archimedean primes, was the Artin-Whaples paper 1945]. Hasse reproduced the proof in the second edition of his textbook \Zahlentheorie". In the rst edition, which had been completed in 1938 already, he proves the approximation theorem for algebraic number elds and algebraic function elds only. More precisely: He rst veri es the theorem for Q and for the rational function eld k (X ) over a eld, and then proves: If the approximation theorem holds for K and its primes p1 ; : : : ; pr then it holds for any nite algebraic extension eld L of K with respect to the nitely many primes of L which are extending some pi . From F. K. Schmidt's arguments in 1933] a proof of the approximation theorem can be extracted for two primes, i.e., r = 2. This is essentially the start of ArtinWhaples' induction proof for arbitrary r. If only non-archimedean primes are concerned, the theorem was formulated and proved by Ostrowski in 1934] already. If one follows Ostrowski's argument carefully then one can see that his proof remains valid if archimedean primes are included. But he did not formulate his result in this generality. Even before Ostrowski's paper was published, Krull in 1930c] had given a proof. Ostrowski cites Krull but observes that Krull formulated only two special cases. This is true but it is clear that from those special cases the general theorem of Ostrowski follows immediately. Deuring in 1931a] did not formulate the approximation theorem in general, but his arguments implicitly contain a proof in the situation which he considers: namely for a nite Galois extension LjK and the nitely many primes P of L which extend a given non-archimedean prime p of K . According to Ostrowski's words, his manuscript for 1934] had been essentially completed in 1916 already, much earlier than the papers by Krull, Deuring, F. K. Schmidt and Artin-Whaples. The essential arguments in all those proofs are similar. So, who should be credited to have been the rst one for proving the approximation theorem ? One should also recall that in the case of the rational number eld, the approximation theorem is essentially equivalent to what is called the \Chinese remainder theorem" which has been known to Sun-Tzi in the 3rd century. Perhaps it is best to keep the name \Artin-Whaples' approximation theorem" which Hasse has used. 4.3 Structure of complete discrete elds. 40 Peter Roquette 4.3.1 The approach of Hasse and F. K. Schmidt. Let us start with a free translation of the rst sentences of the paper 1933c] by Hasse and F. K. Schmidt. In modern algebra one can observe two lines of thought. The rst one is the tendency towards axiomatization and generalization, in order to understand the mathematical phenomena as being part of a general theory which depends on a few simple, far reaching hypotheses only. The second one is the desire to characterize the structures which are formed by those abstract axioms and notions, thus returning from the general to the possible special cases. The rst is often called the abstract or formal point of view while the second can be described as the concrete viewpoint in the framework of modern algebra. The present paper belongs to the second of those ideas. The authors start from the general notion of complete valued eld in the sense of Kurschak, which had evolved from Hensel's investigations of p-adic numbers. Now they are going to characterize all possible complete valued elds, at least in the case of discrete valuations. From these words it appears that the authors regarded their work as being fundamental, and as part of the general development of what at that time was called \modern algebra". The introduction is very elaborate, spanning over 14 pages. The whole paper has 60 pages. 80 Its aim is to present an explicit description of all complete discrete valued elds K . But it was not the last word on the subject. Three years later there appeared the papers by Teichmuller and Witt 1936d], 1937a], 1937b] which contained great simpli cations. Other simpli cations were given by Mac Lane 1939]. Today the paper of Hasse and F. K. Schmidt is almost forgotten, being superseded by those of Witt, Teichmuller and Mac Lane. In particular the so-called Witt vector calculus has obtained universal signi cance. The Hasse-F. K. Schmidt paper had served as a starter for this development. There are two cases to be considered: First, the equal characteristic case, where the characteristic of the complete eld K equals the characteristic of its residue eld K , and secondly the unequal characteristic case, where char(K ) = 0 and char(K ) = p > 0. Hasse-F. K. Schmidt and also Teichmuller try to deal with both cases simultaneously, as far as possible. But here, for better understanding, let us discuss these two cases separately. We begin with equal characteristic case. The cooperation of Hasse and F. K. Schmidt for the paper 1933c] started in early February 1930 when, as reported in section 4.2 already, F. K. Schmidt visited Hasse in Halle. On that occasion Hasse told him that he was working on the classi cation of complete discrete valued elds. After his return to Erlangen F. K. Schmidt asked Hasse, in a letter of Feb 14, 1930, to send him more details. Hasse did so and on Feb 23, F. K. Schmidt thanked him for his \beautiful proof" (schonen Beweis ). Apparently Hasse's proof concerned only the equal characteristic case. For F. K. Schmidt, in his reply of Feb 23, refers to this case only. In addition he observes: Bei der Lekture el mir sogleich auf, da der wesentliche Teil Ihres Satzes von der Voraussetzung, die Bewertung sei diskret, ganz unabhangig ist. 80 This length is partly caused by the fact that the authors include a detailed explanation of the fundamental notions of general valuation theory, starting from Kurschak. By the way, the editing of the text of the paper was done by F. K. Schmidt. This is unusual since Hasse usually was very careful in the wording of his papers and, accordingly, mostly undertook himself the writing and editing of joint papers. History of Valuation Theory, Part I 41 On the rst reading I observed that the essential part of your theorem is independent of the hypothesis that the valuation is discrete. And he proceeded to formulate the statement which he (F. K. Schmidt) thought he was able to prove: If the complete eld K and its residue eld K have the same characteristic then K contains a sub eld K which represents the residue eld K { irrespective of whether the valuation is discrete or not. In today's terminology: Let O denote the valuation ring of K . Then the residue map O ! K admits a section K ?! K O. F. K. Schmidt adds that in order to prove this statement for arbitrary, not necessarily discrete valuations, one has to modify Hasse's arguments in the case of inseparabilities but otherwise Hasse's proof can be used word for word. We do not know Hasse's proof but the above seems to indicate that Hasse considered only the case of residue elds of characteristic zero where there are no inseparabilities. Or, maybe in characteristic p he considered nite residue elds only which of course would be the most interesting case for number theory. Also we do not know what F. K. Schmidt had in mind when he mentioned \modi cations in case of inseparabilities". It seems to us that he mentioned this somewhat in haste but later, in the course of time, he realized that dealing with inseparabilities was more intricate than he originally had thought. For, in a later letter dated March 29, 1930, he mentions brie y that there has appeared a new di culty which however he hopes to overcome. Let us review the situation: The residue eld K is said to be separably generated if there exists a transcendence basis T = (ti ) such that K is separable algebraic over F p (T ). If this is the case then a section K ! O can be constructed as follows: For each element ti of the transcendence basis choose an arbitrary foreimage ti 2 O and let T = (ti ); then the assignment T 7! T de nes an isomorphism F p (T ) ?! F p (T ), hence a section F p (T ) ! O . Because of Hensel's Lemma this has a unique prolongation to a section K ! O. For, every a 2 K is a zero of a monic separable polynomial f (X ) 2 F p (T ) X ]; let f (X ) denote the image of that polynomial in F p (T ) X ]. By Hensel's Lemma there exists a unique foreimage a 2 O of a such that f (a) = 0. The assignment a 7! a (for a 2 K ) then yields a section K ! O. We see: If the residue eld K is separably generated then indeed, there exists a section K ! O of the residue map O ! K , and this does not depend on the structure of the value group. Concerning the extra hypothesis about K being separably generated, we read in the Hasse-F. K. Schmidt paper 1933c]: : : : Das ist im charakteristikgleichen Fall zuerst von dem Alteren von uns Hasse] erkannt worden, daran anschlie end allgemein von dem Jungeren Schmidt], wobei sich uberdies die Moglichkeit ergab, die diskrete Bewertung von K durch eine beliebige Exponentenbewertung zu ersetzen, hinsichtlich der nur K perfekt sein mu . : : : In the case of equal characteristic, this has been discovered rst by the elder of us Hasse], subsequently in general by the younger 42 Peter Roquette Schmidt], whereby it turned out to be possible to replace the discrete valuation of K by an arbitrary non-archimedean valuation, with the only condition that K is complete. This seems to indicate that indeed, Hasse in his letter to F. K. Schmidt had discussed the equal characteristic case only, as we have suspected above already. Moreover we see that F. K. Schmidt now has taken back his former assertion in his letter of Feb 23 to Hasse, and that he has realized that this method works only for separably generated residue elds . Under this extra hypothesis it works not only for discrete valuations, but generally. But perhaps every eld is separably generated? Certainly this is true in characteristic 0; in this case the method of Hasse-F. K. Schmidt is quite satisfactory. In the following discussion let us assume that the characteristic is p > 0. If the eld is nitely generated (over its prime eld) then F. K. Schmidt himself had proved in 1931c] that there exists a separating basis of transcendency. 81 But for arbitrary elds this seemed at rst to be doubtful. In the next letters after Feb 23, 1930 this problem is not mentioned. Only in a later letter, of Sep 14, 1930, F. K. Schmidt returns to this question. In the meantime he had met Hasse in Konigsberg at the meeting of the DMV. 82 They had done a walk together at the beach of the Baltic sea and discussed their joint paper in progress. Now, after F. K. Schmidt's return to Erlangen, he writes: Nun mu ich Ihnen leider doch hinsichtlich der Franzschen Vermutung in der Theorie der unvollkommenen Korper etwas sehr Schmerzliches schreiben : : : Ich kann namlich nun durch ein Beispiel zeigen, da die Franzsche Vermutung tatsachlich bereits bei endlichem Transzendenzgrad nicht zutri t. Genauer: ich kann einen Korper K von Primzahlcharakteristik p angeben mit folgenden Eigenschaften: 1. K ist von endlichem Transzendenzgrad n uber seinem vollkommenem Kern k . 2. Ist t1 ; : : : tn irgendeine Transzendenzbasis von K jk, so ist stets K jk(x1 ; : : : ; xn ) von zweiter Art, wie man auch die Transzendenzbasis wahlen mag. Unfortunately I have to write you something painful with respect to the conjecture of Franz : : : For, I am now able to show by an example that the conjecture of Franz does not hold even for nite degree of transcendency. More precisely: I can construct a eld K of prime characteristic p with the following properties: 1. K is of nite transcendence degree over its perfect kernel k. 2. If t1 ; : : : tn is any transcendence basis of K jk then K jk(x1 ; : : : ; xn ) is always of the second kind, regardless of how one chooses the transcendence basis. 83 From this and other parts of the letter it appears that: 81 Actually, he proved it for transcendence degree 1 only but his proof can be modi ed for arbitrary nite transcendence degree. See, e.g., van der Waerden's textbook 1930e]. 82 DMV = Deutsche Mathematiker Vereinigung (German Mathematical Society). The annual DMV-meeting in Konigsberg was scheduled in the rst week of September, 1930. At that meeting, F. K. Schmidt gave a talk on his results about his uniqueness theorem for complete elds; see section 4.2. Hasse's talk was about the arithmetic in skew elds; see section 3.4.1. 83 F. K. Schmidt still uses the old terminology \ rst kind" and \second kind". In the published paper 1933c] already the new terminology \separable" and \inseparable" is used. History of Valuation Theory, Part I 43 ? ? ? ? ? Franz 84 had written a manuscript in which he conjectured that any eld of nite transcendence degree is separably generated, Hasse had told this to F. K. Schmidt in Konigsberg, F. K. Schmidt had generalized this on the spot to arbitrary transcendence degree, after having returned to Erlangen F. K. Schmidt found an error in Franz' as well as in his own argument, and he got a counter example to Franz' conjecture. But for the residue eld of a discrete complete valued eld, F. K. Schmidt apparently found a way to overcome that di culty. In the published paper 1933c] a method is presented to construct a section K ?! K O, regardless of whether K is separably generated or not. Consequently, if is a prime element of K then it follows that K = K(( )), the power series ring over K. This leads to the following theorem of the Hasse-F. K. Schmidt paper 1933c]. 85 Structure Theorem (equal characteristic case). Every complete discrete valued eld K with char(K ) = char(K ) is isomorphic to the power series eld K ((X )) over its residue eld K . F. K. Schmidt's construction is somewhat elaborate and not easy to check. Indeed, it seems that nobody (including his co-author Hasse) did check the details at the time, and his proof was generally accepted. The rst who really worked on the proof and checked the details seems to have been Mac Lane. In 1939] he pointed out an error in the proof. (This error concerned both the equal and the unequal characteristic case). Mac Lane politely speaks of an \unproven lemma", and he gives a new proof of the Structure Theorem (in both cases), not depending on that lemma. We shall discuss Mac Lane's proof in section 4.3.4 below. Here, let us mention only that the error occurred in the handling of inseparabilities of the residue eld; this concerns precisely the di culty which F. K. Schmidt had mentioned in his letters to Hasse. The \unproven lemma" turned out to be false but a certain weaker result was already su cient to carry through the construction of F. K. Schmidt. Mac Lane planned to write a joint paper with F. K. Schmidt, proving that weaker result and thus correcting the error in 1933c]. Their paper was to appear in the Mathematische 84 Wolfgang Franz was a student of Hasse. He obtained his Ph.D. 1930 at the University of Halle. The subject of his thesis 1931d] was about Hilbertian elds, i.e., elds in which Hilbert's irreducibility theorem holds. One of Franz' results was that a separably generated function eld of one or several variables over a base eld, is Hilbertian. This explains Franz' interest in separably generated elds. The thesis of Franz initiated a new direction of research, namely the investigation of the Hilbert property of elds in connection with valuation theory. (For a survey see e.g., Jarden's book 1986] on Field Arithmetic.) { Franz edited the \Marburger Vorlesungen " of Hasse on class eld theory 1933g]. Later he left number theory and algebra, and started to work in topology. 85 If the residue eld K is nite then the following theorem, as well as the corresponding structure theorem in the unequal characteristic case, is contained in the Groningen thesis of van Dantzig 1931e] already. This is acknowledged in a footnote of the Hasse-F. K. Schmidt paper 1933c]. There the authors say that the essential di culties which appear in their paper do not yet show up if the residue eld is nite. Indeed, from today's viewpoint the structure theorems of Hasse-F. K. Schmidt are almost immediate in case of nite residue eld. But it should be mentioned that the thesis of van Dantzig contained much more than this, it was the rst attempt of a systematic development of topological algebra. 44 Peter Roquette Zeitschrift. 86 But due to the outbreak of world war II their contact was cut o , and so Mac Lane single-handedly wrote a brief report in 1941], naming F. K. Schmidt as co-author and stating the correct lemma which when used in 1933c] saves F. K. Schmidt's construction. Today this note as well as F. K. Schmidt's construction is forgotten since, as said above, it is superseded by the results of Witt and Teichmuller, and later those of Mac Lane. The di culties which F. K. Schmidt faced while struggling with the details of his proof can be measured by the time needed for completion of the manuscript. The authors had started their work on this topic in February 1930, and in November that year F. K. Schmidt sent to Hasse the rst version. But some time later he decided to write a new version. There followed a series of exchanges back and forth of this manuscript. In October 1931 F. K. Schmidt sent the rst part of another version to Hasse who had wished to include the paper into the Crelle volume dedicated to Kurt Hensel. But it was too late for inclusion since the Hensel dedication volume was scheduled to appear in December 1931 already. 87 The published version carries the date of receipt as April 24, 1932. But even after that date there were substantial changes done. In a letter dated July 11, 1932 F. K. Schmidt announced that now the whole paper has to be rewritten. And even on April 4, 1933 F. K. Schmidt inquired whether it is still possible to make some changes in the manuscript. Finally, in an undated letter (written in May or June 1933) F. K. Schmidt gave instructions for the last and nal changes while Hasse was reading the galley proofs of the manuscript. 4.3.2 The Teichmuller character. We are now going to describe Teichmuller's beautiful construction in 1936d] which superseded the Hasse-F. K. Schmidt construction. Consider rst the case when the residue eld K is perfect. Its characteristic is a prime number p . Given an element in the residue eld?K , Teichmuller chooses, for n = 0; 1; 2; : : : , an arbitrary representative an of p n . Then each apn is a n representative of the given . Teichmuller 1936d] observed that n (22) ( ) := nlim ap !1 n exists and is independent of the choice of the an . (Here he used the fact that the valuation is discrete.) It is seen that ( ) = ( ) ( ): (23) The multiplicative map : K ! O is called the Teichmuller character ; its image R = (K ) is the Teichmuller representative set for the residue map. It is the only representative set for which Rp = R. This construction works in both cases, the equal characteristic case and the unequal characteristic case. In the equal characteristic case we have besides of (23) the corresponding formula for the addition: ( + ) = ( )+ ( ): (24) 86 They had discussed this plan when F. K. Schmidt visited the United States in 1939 as a representative of the Springer Verlag, in connection with the American plans to establish Mathematical Reviews . See 1979a]. { It is to be assumed that Mac Lane and F. K. Schmidt had met earlier between 1931 and 1933, when Mac Lane studied in Gottingen. 87 It seems that F. K. Schmidt was not unhappy with this outcome because he wished to make still other changes. Hasse, on the other hand, had in the meantime (November 9, 1931) succeeded, together with R. Brauer and E. Noether, to prove the Local-Global Principle for algebras, and so he put that new manuscript into the Hensel dedication volume instead. History of Valuation Theory, Part I 45 Therefore R is a eld and : K ?! R is a section. The beauty of this construction is not only its simplicity but also that it is canonical. If K is not perfect then Teichmuller uses the theory of p-bases of K which he had introduced in a recent paper in another context 1936e]. Choose a p-basis nM of K and a foreimage M O ; then consider the union of the elds K (M p? ) , (n = 0; 1; 2; : : : ). Its completion L is a discrete valued complete eld whose residue eld L is the perfect hull of K . This then allows to reduce the problem to the case of perfect residue elds. More precisely: The Teichmuller character : L ! L maps M into M and K into K . In 1937b] Teichmuller gives a more detailed (and in some points corrected) version of this construction. Note that the construction of this section K ! K is not canonical but depends on the choice of the foreimage M of a p-basis of K . Nevertheless it is quite simple and straightforward when compared with the HasseF. K. Schmidt construction 1933c], even when the latter is taken in its corrected form 1941]. 4.3.3 Witt vectors. Next we consider the unequal characteristic case, which means that char(K ) = 0 and char(K ) = p > 0. In this situation K is called unrami ed if p is a prime element of K . The main result of Hasse-F. K. Schmidt for unrami ed elds is as follows: Structure Theorem (unequal characteristic case) An unramied, complete discrete eld K of characteristic 0 and residue characteristic p > 0 is uniquely determined by its residue eld K (up to isomorphism). For any given eld in characteristic p there exists an unrami ed, complete discrete eld whose residue eld is isomorphic to the given one. Again, the proof given by Hasse-F. K. Schmidt in 1933c] is quite elaborate and not easy to check. The error which Mac Lane had pointed out in 1939] and corrected in 1941], concerns the unequal characteristic case too. We now discuss the papers 1936e] and 1937] by Teichmuller and by Witt which greatly simpli ed the situation. Similar as in the equal characteristic case, suppose rst that K is perfect, so that the Teichmuller character : K ! O is de ned as in (22), with the multiplicative property (23). (Recall that O denotes the valuation ring of K .) Again let R = (K ). Since we are discussing the unrami ed case, p is a prime element of K and therefore every element a 2 O admits a unique expansion into a series with a0n 2 R : a = a00 + a01 p + a02 p2 + Since R = Rp there are an 2 R with a0n = apn . We obtain an expansion of the n form 2 with an 2 R : (25) a = a 0 + ap p + a p p 2 + 2 1 Let an correspond to n 2 K via the Teichmuller character, i.e., an = ( n ). We see that a is uniquely determined by the vector W (a) = ( 0 ; 1 ; 2 ; : : : ) (26) with components n in the residue eld K . 46 Peter Roquette There arises the question how the components of W (a b) are computed from the components of W (a) and W (b). And similar for W (a b). Teichmuller 1936e] and also H. L. Schmid 1937c] had discovered that this computation proceeds via certain polynomials with coe cients in Z, not depending on the residue eld K but only on p, its characteristic. At this point the discovery of Witt becomes crucial. He could show that those polynomials can be obtained by a universal algorithm which is easy to describe and easy to work with. Thus Witt gave an explicit, canonical construction of a complete discrete valuation ring W (K ) of characteristic 0, based on the given eld K of characteristic p, and he showed that O W (K ). 88 In my opinion, in order to give a brief description of the algorithms of Witt vectors, the best way to do so is to cite his words which he himself used in his seminal paper 1937a]. We do this in a free English translation: For a vector x = (x0 ; x1 ; x2 ; : : : ) with countably many components xn we introduce ghost components 89 n pn?1 x(n) = xp + p x1 + + pn xn : 0 Inversely, xn can be expressed as a polynomial in x(0) ; x(1) ; : : : ; x(n) with rational coe cients, and therefore the vector x is already determined by its ghost components. We de ne Addition, Subtraction and Multiplication of two vectors via the ghost components: (x y)(n) = x(n) y(n) (n = 0; 1; 2; : : : ): Satz 1. (x y)n is a polynomial in x0 ; x1 ; : : : ; xn ; y0 ; y1 ; : : : ; yn with integer coe cients . In view of his \Satz 1" Witt observes that one can substitute for the indeterminates x0 ; x1 ; : : : ; y0 ; y1 ; : : : elements of an arbitrary ring A and obtain a new ring W (A), the ring of Witt vectors over A. In particular for A = K we obtain W (K ). Witt proves this to be a complete discrete valuation ring of characteristic 0, with p as prime element. Witt shows further that the map a 7! W (a) de ned by (25), (26) is an isomorphism of the valuation ring O of K with W (K ). Thus: Witt's Theorem: Every unrami ed complete discrete valuation ring O of characteristic 0 with perfect residue eld K of characteristic p > 0 is isomorphic to the Witt vector ring W (K ). Hence the quotient eld K of O is isomorphic to the quotient eld QW (K ) of W (K ). We see that QW (K ) plays the same role in the unequal characteristic case, as the power series eld K ((X )) does in the equal characteristic case. If the residue eld K is not perfect then W (K ) is not a valuation ring. But Teichmuller in 1937b] was able to prove the structure theorem also in this case, with the help of a construction using a p-basis of K , similarly as in the equal characteristic case. 88 Perhaps it is not super uous to point out that the ring W (K ) of Witt vectors has nothing to do with what is called the \Witt ring" of the eld K which is de ned in the theory of quadratic forms. 89 Witt says \Nebenkomponenten". 47 History of Valuation Theory, Part I Finally, if K is rami ed (in the unequal characteristic case) then it is shown that QW (K ) is canonically contained in K , and that K is a nite algebraic, purely rami ed extension of QW (K ), and that it can be generated by a root of an Eisenstein equation. This is routinely derived from what we have seen above. Remark 1. The calculus of Witt vectors has implications throughout mathematics, not only in the construction of complete discrete valuation rings. In fact, it was rst discovered by Witt in connection to another problem, namely the generalization of the Artin-Schreier theory to cyclic eld extensions of characteristic p and degree pn . This is explained in Witt's paper 1937a]. Later, Witt generalized his vector calculus such as to become truly universal, i.e., not referring to one particular prime number p. Instead of considering the powers p0 ; p1 ; p2 ; : : : of one prime number p, he considers those of all prime numbers and their products, which is to say the natural numbers 1; 2; 3; : : : . The ghost components of a vector x = (x1 ; x2 ; x3 ; : : : ) are now de ned to be X x(n) = d xn=d : d djn This is a truly universal calculus, and the Witt vectors in the sense of his original paper 1937a] represent, in a way, only a certain part (not a subring!) of the universal Witt vectors. See, e.g., Witt's article which is published posthumously in his Collected Works 1998f] and the essay of Harder therein. Certainly, the discovery of Witt vectors is to be rated as one of the highlights of mathematics in the 20th century { independent of the application to valuation theory which we have discussed here. Remark 2. Both the papers of Witt 1937] and of Teichmuller 1937b] were published in the same issue of Crelle's Journal, namely Heft 3, Band 176 . 90 This whole issue consists of 8 papers by Witt, Teichmuller, H. L. Schmid and Hasse. The latter, as the editor of Crelle's Journal, had opened this issue for the Gottinger Arbeitsgemeinschaft (workshop) in which, as we can see, remarkable results had been achieved. 91 4.3.4 Mac Lane. Saunders Mac Lane (1909{) studied from 1931 to 1933 in Gottingen where he came in contact with \modern" mathematics and algebra. His Ph.D. thesis was written under the supervision of Bernays, on a topic from mathematical logic. 92 Here we are concerned with his contributions to valuation theory. Kaplansky tells us in 1979a] that Mac Lane's interest in valuation theory can be traced quite directly to the in uence of Oystein Ore at Yale, where Mac Lane had studied in 1929/1930 and to where he returned in 1933/1934 after his Gottingen interlude. 90 At that time, a volume of Crelle's Journal consisted of 4 issues (4 Hefte ). 91 The names of the members of the Arbeitsgemeinschaft in the years 1935/36 are mentioned in 1936d]. The two most outstanding members (besides of Hasse) were Teichmuller and Witt. In a letter of Hasse to Albert dated Feb 2, 1935, Hasse mentioned \Dr. Witt, our best man here ". About Teichmuller he wrote to Davenport on Feb 3, 1936, that he (Hasse) was working with Teichmuller on cyclic elds of degree p. And he added: \His paper on the Wachs-Raum is as queer as the whole chap." 92 Biographical information about Mac Lane, who became one of the leading mathematicians in the United States, can be found in his \Selected Papers" 1979b]. 48 Peter Roquette Already in section 4.3.1 we have mentioned the paper 1939] of Mac Lane where it was pointed out that the Hasse-F. K. Schmidt paper 1933c] contained an error, and where a new, correct proof of the Structure Theorem of Hasse-F. K. Schmidt was presented. But why was it necessary in the year 1939 to give a new proof ? After all, a beautiful new proof had already been given in 1937 by Teichmuller and Witt, as reported in the foregoing sections 4.3.2 and 4.3.3. Mac Lane mentions that Witt 1937a] uses a \sophisticated vector analysis construction "; he wants to avoid Witt vectors and prove the Structure Theorem by more elementary means. He does not attempt, however, to avoid the use of the Teichmuller character; it is only the Witt vectors which he wishes to circumvent. Hence Mac Lane discusses the unequal characteristic case only, since the equal characteristic case had been solved by Teichmuller without Witt vectors, as reported in section 4.3.2. Mac Lane's new idea is to divide the proof into two steps: rst the existence proof, i.e., the construction of a discrete valued unrami ed complete eld of characteristic 0 with a given residue eld of characteristic p ; second, the uniqueness proof, i.e., the construction of an isomorphism between two such elds with the same residue eld. The separation of these two steps shows, Mac Lane says, \that the previous constructions he means those by Witt] have been needlessly involved and can be replaced by an elementary stepwise construction ". The existence part is easily dealt with: the given eld of characteristic p which is to become the residue eld, is an extension of the prime eld F p of characteristic p which in turn is the residue eld of Q p . Therefore one has to verify the following general Relative Existence Theorem. Let k be a non-archimedean complete valued eld with residue eld k, and let K be an extension eld of k. Then there exists a complete unrami ed extension K of k whose residue eld is the given K . By Zorn's Lemma 93 it su ces to discuss the case when K = k( ) is a simple extension. If is algebraic over k let f (X ) be the monic irreducible polynomial for over k, and let f (X ) 2 k X ] be a monic foreimage of f (X ). Then f (X ) is irreducible over k. If a is a root of f (X ) then the eld K = k(a) solves the problem; the valuation of k extends uniquely to K and K is complete by the very rst results of Kurschak and Ostrowski. Moreover, K jk is unrami ed since K : k] = K : k]. This holds regardless of whether f (X ) is separable or not. On the other hand, if is transcendental over k then consider the rational function eld k(x) in an indeterminate x over k, with its functional valuation. The completion K of k(x) then solves the problem. We see that this existence part holds not only for discrete valuations; the arguments are valid quite generally and they do not give any problem with respect to inseparabilities of the residue eld. And they were well known even at the time of Mac Lane. Thus the main point of Mac Lane's paper is the second part about uniqueness. Here he shows the following: Relative Uniqueness Theorem. Let k be a complete valued eld and assume that the valuation is discrete. Let K and K 0 be two 93 Mac Lane does not yet use Zorn's Lemma of 1935b]; he refers to \known" methods using well orderings. History of Valuation Theory, Part I 49 complete unrami ed extensions of k. Suppose that there exists a kisomorphism of the residue elds K ! K 0 . Then this can be lifted to a k-isomorphism K ! K 0 { provided K is separable over k. Here, the essential new notion was that of \separable" eld extension without the assumption that K is algebraic over k. Mac Lane had discovered that for an arbitrary eld extension K jk of characteristic p , the adequate notion of \separable" is not the one used by Hasse and F. K. Schmidt (which was \separably generated ") but it should be de ned as \k-linear disjoint to k1=p ". Equivalently, this means that p-independent elements of k (in the sense of Teichmuller 1936e]) remain pindependent in K . Mac Lane in 1939a] was the rst one to study this notion systematically and pointing out its usefulness in dealing with eld extensions of characteristic p . If there exists a separating transcendence basis then K jk is separable in the above sense, but the converse is not generally true. Every eld in characteristic p is separable over its prime eld F p { and this, as Mac Lane points out, is the fact which allows to deduce the \absolute" uniqueness theorem which says that an unrami ed complete eld is uniquely determined by its residue eld. If the residue eld K is perfect then the proof of the relative uniqueness theorem is almost immediate when the Teichmuller character : K ! K is used. For, choose a basis of transcendency T of K jk, and put T = (T ). Similarly, after identifying K = K 0 by means of the given isomorphism, put T 0 = 0 (T ) where 0 : K ! K 0 is the Teichmuller character for K 0 . Then the map T 7! T 0 yields an isomorphism of k(T ) onto k(T 0 ) as valued elds. Since K is perfect, every t 2 T has a unique p-th power in K , hence (T 1=p ) = T 1=p is well de ned in K ; similarly 0 (T 1=p ) = T 01=p in K 0 . In this way the above isomorphism extends uniquely to an isomorphism k(T 1=p ) ! k(T 01=p ): Similarly for k(T 1=p2 ); k(T 1=p3 ); : : : , then to the union of these elds, and then to its completion. Further extensions involve only separably algebraic extensions of the residue eld and hence can be dealt with by Hensel's Lemma. If the residue eld K is not perfect then Mac Lane uses the Teichmuller construction for imperfect residue elds as given in 1936d]. The above discussion shows that the main merit of Mac Lane's paper is not so much a new proof of the Witt-Teichmuller theorem about complete discrete valued elds in the unequal characteristic case. 94 The speci c and highly interesting description of those elds by means of the Witt vector calculus is completely lost in Mac Lane's setup. In our opinion, the main point of Mac Lane's paper is the discovery of the new notion of \separable eld extension" which indeed became an important tool in eld theory in characteristic p. It is not only applicable in the proof of Mac Lane's Uniqueness Theorem but it has proved to be useful in many other situations, e.g., in algebraic geometry of characteristic p. In this respect Mac Lane's paper 1939] and the follow up 1939a] are indeed to be called a \classic", as does Kaplansky in 1979a]. Moreover, Mac Lane's method of describing the structure of discrete complete valuation rings has since served as a model for further similar descriptions in topological algebra where Witt vectors cannot be used. See, e.g., Cohen's paper 1946a] on the structure of complete regular rings, as well as Samuel 1953], Geddes 1954], and also 1959]. 94 Mac Lane calls this \the p-adic case". 50 Peter Roquette Let us note a minor but curious observation. In his paper 1939] Mac Lane mentions that the Hasse-F. K. Schmidt paper 1933c] contains an error and he sets out to give a new, correct proof. The proof is correct but a certain more general statement is not. Consider the situation of the relative uniqueness theorem. If the residue eld K admits a separating basis of transcendency then, Mac Lane claims, the conclusion of that theorem holds even without the hypothesis that the valuation of k is discrete. Certainly this is not generally true in view of the existence of immediate extensions of complete elds with non-discrete valuations. In a subsequent paper 1940] he corrects this error, mentioning that his student I. Kaplansky had called his attention to the counter examples. We note that Mac Lane's error is of the same type as F. K. Schmidt's in his letter to Hasse of Feb 23, 1930 ! (See section 4.3.1.) 5 Ostrowski's second contribution We will now discuss Ostroswki's great paper \Untersuchungen zur arithmetischen Theorie der Korper " 1934]. This paper has 136 pages; it almost looks like a monograph on valuation theory. It seems to be written as a follow up to Ostrowski's rst papers 1913a], 1917], 1918]. Indeed, Ostrowski tells us in the introduction that a substantial part of its contents was already nished in the years 1915-1917. This was even earlier than Rychl k's 1919]. And we shall see that the Ostrowski paper contains very important and seminal ideas. I do not know why Ostrowski waited more than 14 years with the publication of these results. The author says it was because of the adversity of the times and the great length of the manuscript (\die Ungunst der Verhaltnisse und wegen des gro en Umfangs von 110 Manuskriptseiten "). Perhaps, we may guess, his interest had partly shifted to other problems in mathematics, for within those 14 years he published about 70 other papers of which only a few can be said to have been in uenced by valuation theory, and only one had a closer connection to valuation theory proper, namely 1933a] on Dirichlet series. Ostrowski's former valuation theory papers were written and published during the years of the rst world war 1914{1918. We are told by Jeltsch-Fricker 95 in 1988] that those years were very special for Ostrowski who, as said above, studied in Marburg (Germany) at that time. Since he was a Russian citizen and Russia was at war with Germany, he was con ned to internment. On the intervention of Kurt Hensel however, he was granted certain privileges, among them the use of the university library at Marburg. In later years Ostrowski said that this outcome had been reasonably satisfactory for him and he did not consider the four years of the war as wasted, for: The isolation enabled him to concentrate fully on his investigations. He read through mathematical journals, in his own words, from cover to cover, occupied himself with the study of foreign languages, music and valuation theory (almost completely on his own). 96 We see that valuation theory is explicitly mentioned. On the other hand, it is also mentioned that he had learned a lot more by reading mathematical journals 95 She had been assistant to Ostrowski in Basel. 96 Cited from Jeltsch-Fricker 1988]. History of Valuation Theory, Part I 51 from cover to cover 97 , and hence he may have discovered that his interests were not con ned to valuation theory. In fact, after the war was ended and he was free again to move around in Germany, he left Marburg and went to Gottingen. 98 There he was absorbed with quite di erent activities. He obtained his Ph.D. in the year 1920 with E. Landau and D. Hilbert. The title of his thesis was \Uber Dirichletsche Reihen und algebraische Di erentialgleichungen." 99 We suspect that by 1932 Ostrowski realized that valuation theory had developed considerably in the meantime, and that a number of published results (e.g., by Deuring and by Krull) were contained as special cases in his old, unpublished manuscript. But perhaps it was his work on Dirichlet series 1933a] which had prompted him to dive into valuation theory again; in that paper he had used some facts about Newton diagrams for poynomials over Henselian elds. Moreover, there had appeared Krull's fundamental paper 1932g] which contained much of the basic ideas of Ostrowski concerning what he calls the abstract \Riemann surface" of a eld. In any case he decided to nally publish his old manuscript, enriched with certain new ideas concerning the abstract Riemann surface which he dates to April{July 1932. He submitted the manuscript for publication to the \Mathematische Annalen" in 1932, and it was published in 1934. \Ostrowski's Theorem" in 1917] (see section 2.2.3) had stated that every archimedean valued eld is isomorphic, as a topological eld, to a sub eld of the complex number eld C . (See section 2.2.3.) In view of this result, his new paper 1934] is devoted exclusively to non-archimedean valuations. Instead of valuations he also speaks of prime divisors of a eld K { a notion which, as we have seen above, he had already introduced in his earlier paper 1918]. He uses valuations in the additive form; thus a prime divisor p of the eld K is given by a map v : K ! R +1 such that v(a) < +1 if a 6= 0, and v(0) = +1 (27) v(a + b) min(v(a); v(b)) (28) v(ab) = v(a) + v(b) (29) 9a : v(a) 6= 0; +1 (30) Ostrowski calls v(a) the \order" (Ordnungszahl ) of a at the prime p. Through this notation and terminology he refers to the analogies from the theory of complex functions and Riemann surfaces { a viewpoint which penetrates the whole of Ostrowski's theory in this paper. It seems that Ostrowski was guided to a large extent by his experiences with the handling of Dirichlet series in his former papers. 97 It is reported that Ostrowski had a phenomenal memory capacity. In later years, when he was studying in Gottingen, he was used by his fellow students as a living encyclopedia. If one wanted to know details about when and where a particular mathematical problem had been discussed and who had done it, one could ask Ostrowski and would obtain the correct answer, including the volume number and perhaps also the page number of the respective articles. 98 There is a postcard, dated June 12, 1918, from Emmy Noether in Gottingen to Ostrowski in Marburg, expressing her delight that he will soon move to Gottingen. The question arises whether Ostrowski had shown his valuation theoretic manuscript of 1934], which he said to have been essentially completed in 1917 already, to Emmy Noether. But there is no evidence of this, and we believe he rather did not . Otherwise she would probably have told her student Deuring about it, 12 years later when Deuring wrote his paper 1931a] (see section 4.1.1). After all, Deuring's results were contained in Ostrowski's (see section 5.2). But Deuring does not cite Ostrowski's manuscript, and so we believe that he did not know about it. 99 This was connected with one of Hilbert's problems of 1900, and hence stirred much interest. 52 Peter Roquette In fact, the rst example of valued elds which Ostrowski presents here are elds of Dirichlet series. According to the de nitions, the valuation v is uniquely determined by its prime divisor p up to equivalence, which means up to a real positive factor. Ostrowski's paper consists of three parts. Each part is rich in interesting results, and we discuss them in three separate sections. 5.1 Part I. Henselian elds. After the basic de nitions and facts of valuation theory, Ostrowski introduces what today is called a \Henselian eld", i.e., a valued eld K in which Hensel's Lemma holds. In this connection it is irrelevant which of the various equivalent forms of Hensel's Lemma is used in the de nition. Ostrowski uses the following: Fundamental Lemma: Consider a polynomial f (x) = an xn + an?1 xn?1 + + a0 whose coe cients belong to the valuation ring of the prime p of K . Assume that f (x) an xn + + am+1 xm+1 + xm mod p with n > m > 0 where the congruence is to be understood coe cient-wise modulo the maximal ideal of the valuation ring. Then f (x) is reducible in K . If this Fundamental Lemma holds for a valued eld K then Ostrowski calls the eld \relatively complete" (relativ perfekt ). Today the terminology is \Henselian". For the convenience of the reader we prefer to use today's terminology; thus we will always say \Henselian" when Ostrowski says \relatively complete". 100 Apart from the terminology, Ostrowski treats these elds in the same way as we would do today. He clearly sees that the important algebraic property of complete valued elds is not their completeness but the validity of Hensel's Lemma, and he creates his theory accordingly. Ostrowski shows that every valued eld K admits a Henselization K h , unique up to isomorphism, and he proves the standard and well known properties of it. He does this in several steps. b Step 1. It is proved that the completion K is Henselian, which is to say that b the above mentioned Fundamental Lemma holds in K . Ostrowski remarks that this is a special case of the more general reducibility theorem in Rychl k's paper 1923f], but since he does not go for the full Hensel-Rychl k Lemma the proof of the Fundamental Lemma is particularly simple. Step 2. It is proved that for any Henselian eld, its valuation can be extended uniquely to the algebraic closure. Thus Ostrowski repeats a third time what Kurschak 1912] and Rychl k 1923f] had already proved. But the emphasis here is that the result depends on the Henselian property only, and completeness is not required. Of course, this could have been seen immediately from the proof given by Kurschak, or that by Rychl k { provided the notion of \Henselian eld" would have been established. But neither Kurschak nor Rychl k had conceived that notion, and it was Ostrowski who introduced it. 100 We do not know when and by whom the name \Henselian eld" had been coined; probably it was much later. Schilling in his book 1950] still uses the terminology of Ostrowski. Azumaya 1951] introduces the notion of \Henselian ring" in the context of local rings, but he does not tell whether this terminology had been used before for valuation rings or valued elds. History of Valuation Theory, Part I 53 At the same time, the important fact is proved that any algebraic extension LjK , nite or in nite, of a Henselian eld K is Henselian again. (Recall that for extensions of in nite degree, this not true if one replaces \Henselian" by \complete"; this had been shown by Ostrowski in his rst paper 1913a] in reply to Kurschak's question.) Moreover, given any valued eld L then the intersection of two valued Henselian sub elds of L is Henselian again. Step 3. Now, for an arbitrary valued eld K , Ostrowski constructs its Henselizab tion K h as the separable-algebraic closure of K within the completion K . It is h , if de ned in this way, is Henselian and, moreover, that every proved that K Henselian eld extension of K contains a K -isomorphic image of K h . Step 4. At the same time, Ostrowski connects this notion with Galois theory, as follows. He presents a description of the prolongations of the valuation v of K to an arbitrary algebraic extension L of K , nite or not, in the following manner. Those prolongations are all given by the K -isomorphisms 101 of L into the algebraic b closure of the completion K , observing that the image is uniquely valued by Steps 1 and 2. If LjK is a Galois extension 102 then it follows, after xing one prolongation w of v to L, that all other prolongations are obtained from w by the automorphisms of the Galois group. In fact, the other prolongations correspond to the cosets of the Galois group G modulo the decomposition subgroup Z , consisting of those automorphisms which leave w xed. And the xed eld of Z in L is precisely the intersection K h \ L { more precisely, it is isomorphic to it as valued eld. Thus, K h \ L is the decomposition eld of v in L (unique up to conjugates over K ). Taking L = K s to be the separable algebraic closure, it follows that K h is the decomposition eld of v in K s . This explains the terminology of Ostrowski who does not speak of \Henselization" but calls K h the \reduced universal decomposition eld ". The word \reduced" refers to the fact that K h is separable over K , and \universal" means that for any Galois extension LjK the intersection K h \ L is the decomposition eld. All this is quite the standard procedure today. 103 We have mentioned it in such detail because it seems remarkable that these ideas can be found in Ostrowski's paper 1934] which, according to the author, had been written in its essential parts in 1915 already. The main development of valuation theory before Ostrowski's 1934] was in uenced by the applications to number elds, in which case an important role was played by the completion because it permitted to use analytic arguments in the pursuit of arithmetical problems { for instance, the use of exponential and logarithmic function, of the exponentiation with p-adic integer exponents etc. Thus from the arithmetic viewpoint, there was more interest, in those years, in the analytic properties of valued elds which implies working with completions , rather than with Henselizations . The Henselization is the proper notion if one concentrates on the algebraic properties of valuations. With this background, the essential ideas of Ostrowski's paper were perhaps not adequately appreciated by the mathematical public of that time, at least not by those coming from number theory. In fact, in the Zentralblatt review about 101 Ostrowski uses the word \permutation" instead of \isomorphism", thereby following the old terminology of Dedekind. 102 Ostrowski also considers what he calls a \normal extension", which is a Galois extension followed by some purely inseparable extension. 103 Except perhaps that in Step 1 one would today avoid the completion altogether by showing directly that the \reduced universal decomposition eld" (see Step 4) is Henselian. 54 Peter Roquette Ostrowski's paper, F. K. Schmidt did not even mention the notion of Henselian (or relatively complete) eld and says about Part I: \Seine Ergebnisse sind im wesentlichen bekannt.". This indicates that his attention was xed on the completion which he was used to, and he did not appreciate the importance of the new notion of Henselization. 104 In view of this we nd Ostrowski's algebraic theory of Henselizations remarkable indeed. In later times the notion of Henselization turned out to be even more important in the theory of general Krull valuations. In that case the completion is not necessarily Henselian and hence not useful for discussing the prolongation of valuations. Thus Ostrowski, although his paper is concerned with valuations of rank 1 only, prepares the way for the discussion of higher rank valuations too by creating the notion of Henselization and pointing out its importance. This Part I contains the Approximation Theorem for nitely many non-archimedean primes of a eld K . We have discussed this theorem in section 4.2.1. Ostrowski speaks of \independence" (Unabhangigkeit ) of nitely many primes in a eld. 5.1.1 Newton diagrams. In some of his arguments, Ostrowski uses the method of Newton diagram of a polynomial over a valued eld K . He says: If the eld K is Henselian then the sides of the Newton diagram correspond to the irreducible factors of the polynomial. This last statement can be used as the de nition of Hensel elds. The use of the graphic Newton diagram method is not really necessary but it is lucid and the arguments become brief. Ostrowski takes the main statements about Newton diagrams for granted; he does not even give the de nition of the Newton diagram of a polynomial and cites his earlier paper 1933a]. Although there he discusses complete elds only, it is clear from the context that the theory of Newton diagrams of 1933a] applies in the same way to arbitrary Henselian elds in his present paper. It seems that this is the main aim of Ostrowski when he discusses Newton diagrams in his paper. He wishes to put into evidence that these can be useful over arbitrary Henselian elds. 105 5.2 Part II: Rami cation and defect. And he sets out to show this in the second part of 1934]. There he discusses an algebraic extension L of a valued eld K . One of his main results, striking because of its generality, is the following relation, valid for an arbitrary algebraic extension eld L of nite degree n. There are only nitely many extensions of the given valuation of K to L; let r denote their number. For the i-th extension let ei denote the corresponding rami cation index and fi the residue degree. With this notation, Ostrowski proves the important degree relation X n= (31) i ei fi 1 ir where i denotes the \defect" of the i-th extension. In the case of classical algebraic number theory, where the non-archimedean valuations are de ned by prime ideals of Dedekind rings and algebraic extensions are separable, the relation (31) was well 104 The review appeared in vol. 5 of the Zentralblatt. 105 Newton diagrams, in the framework of valuation theory, had also been studied by Rella 1927d] . History of Valuation Theory, Part I 55 known at the time (with trivial defects i = 1) through the work of Dedekind, Hilbert, Weber and, later, Emmy Noether. In the general case it was known that P n i ei fi (as far as this question had been studied). The achievement of Ostrowski was that he could de ne the defect i as a local invariant (i.e., depending only on the local extension Lh jK h of the respective Henselizations) and show that i it is always a power of the characteristic exponent p of the residue eld. For Galois extensions, all the prerequisites of de ning the defect and deriving the formula (31) were contained already in the papers by Deuring 1931a] and Krull 1930c], as we had pointed out already in section (4.1.1). But neither of them took the step to introduce the defect as an invariant which is worthwhile to investigate. Perhaps this can be explained by the fact that in the classical cases in number theory the defect is always trivial and, hence, the appearance of a non-trivial defect was considered to be some pathological situation which was of no particular interest. But not for Ostrowski. He goes to great length to investigate the structure of the defect; in particular to give criteria for i = 1, in which case the extension Lh jK h is defectless . Today, Ostrowski's notion of defect and its generalizations i have become important in questions of algebraic geometry; see, e.g., Kuhlmann's thesis 1989a], and also his paper 2000a]. Ostrowski derives in detail a generalization of a good part of the DedekindHilbert rami cation theory: he de nes and studies decomposition eld, inertia eld, rami cation eld etc. and their corresponding automorphism groups. But Galois theory does not play a dominant role in this paper; mostly the author prefers to consider elds instead of their Galois groups. 106 The reason for this is that he wishes to include, in a natural way, the inseparable eld extensions which cannot be handled by Galois theory. Nevertheless he cites the recent papers by Krull 1930c], Deuring 1931a], and Herbrand 1932b] where similar questions are discussed in a di erent way, relying on Galois theory and ideal theory. (See section 4.1.) As to ideal theory, Ostrowski says explicitly: Wir haben vom Idealbegri keinen Gebrauch gemacht, da es vielleicht einer der Hauptvorzuge der hier dargestellten Theorie ist, da durch sie der Idealbegri eliminierbar wird. We have not made use of the notion of ideal since it is perhaps the main advantage of the theory as presented here, that the notion of ideal can be avoided. Indeed, valuation theory had been created by Kurschak, following Hensel's ideas, in order to have a directly applicable tool to measure eld elements with respect to their size or their divisibility properties, and in this respect to be free from ring or ideal theory. 5.3 Part III: The general valuation problem. 5.3.1 Pseudo-Cauchy sequences. In this part, Ostrowski rst considers the problem of extending a valuation of a eld K to a purely transcendental eld extension K (x). To this end he develops his theory of pseudo-Cauchy sequences 107 106 He even avoids the use of Sylow's theorem from group theory and, instead, uses a trick (\Kunstgri ") on algebraic equations, attributed to Fonceneux. 107 Ostrowski uses the terminology \pseudo convergent" instead of \pseudo-Cauchy". This re ects the terminology in analysis of that time. What today is called \Cauchy sequence" in analysis, was at that time often called \convergent sequence" (meaning convergent in some larger 56 Peter Roquette a1 ; a2 ; a3 ; : : : with ai 2 K ; the de ning property is that nally (i.e., for all su ciently large n) we have either kan+1 ? an k < kan ? an?1 k (32) or an+1 = an : (Here, Ostrowski switches to the multiplicative form of valuations in the sense of Kurschak.) If an is a pseudo-Cauchy sequence in K , or in some valued algebraic extension of K , and 0 6= f (x) 2 K (x) then f (an ) is also a pseudo-Cauchy sequence. Moreover, limn!1 kf (an )k exists and is called the limit value (Grenzbewertung ) of f (x) with respect to the pseudo-Cauchy sequence an . If this limit value does not vanish for all non-zero polynomials then Ostrowski puts kf (x)k = nlim kf (an )k !1 and obtains a valuation of K (x) extending the given valuation in K . In this situation x is called a pseudo limit of the sequence an . Moreover, Ostrowski shows: Every extension of the valuation of K to K (x) is obtained in this way by a suitable pseudo-Cauchy sequence in the valued algebraic closure of K , so that x becomes a pseudo limit of the sequence. This is a remarkable result indeed. The question of how the valuation of K extends to a purely transcendental extension is quite natural and certainly had been asked by a number of people. Traditionally there had been the solution kf (x)k = kc0 + c1 x + + cn xn k = max(kc0 k; : : : ; kcn k) which is called the \functional valuation" of the rational number eld, with respect to the given valuation of the base eld K . More generally, there was known the solution kf (x)k = max(kc0 k; kc1 k ; : : : ; : : : ; kcn k n ) if is any given positive real number; this is called by Ostrowski the extension \by an invariable element". These solutions are now contained as special cases in Ostrowski's construction, namely, if the pseudo-Cauchy sequence an is \of second kind" which means that nally kan+1 k < kan k; we then have = limn!1 kan k. Ostrowski investigates in some detail how the various properties of the pseudoCauchy sequence in uences the properties of the extended valuation. Of particular interest is the case when both the residue eld and the value group are preserved, i.e., when K (x) becomes an immediate extension of K . If K is algebraically closed then this is the case if x is a \proper" pseudo limit of the sequence an . This is de ned by the condition that the distance of x to K is not assumed, i.e., to every b 2 K there exists c 2 K such that kx ? ck < kx ? bk. Equivalently, kan ? ck < kan ? bk for su ciently large n. The notion of pseudo-Cauchy sequence and its use for extending valuations to the transcendental extension K (x) was a completely new idea which perhaps originated in Ostrowski's work on elds of Dirichlet series 1933a]. Anyhow, this aspect of valuation theory went far beyond what was known and studied in classical algebraic number theory where discrete valuations dominated. The introduction and study of pseudo-Cauchy sequences and their pseudo limits is to be viewed as a milestone in general valuation theory. It opened the path to the more detailed study space). Alternatively, the terminology \fundamental sequence" was used, following G. Cantor 1883], p.567. History of Valuation Theory, Part I 57 of Krull valuations of which the well known Harvard thesis of Irving Kaplansky 1942] was to be the rst instance. 5.3.2 Remarks on valuations of rational function elds. The problem of determining all possible extensions of a valuation from K to the rational function eld K (x) had been studied already by F. K. Schmidt. In a letter to Hasse dated March 7, 1930 he investigated all unrami ed extensions of the valuation to K (x). In doing so he answered a question put to him by Hasse. He came to the conclusion that, if the residue eld K is algebraically closed then every unrami ed extension of the valuation is a functional extension with respect to a suitable generator of K (x), and conversely. Of particular interest is a postscript to his letter where he drops the hypothesis that K is algebraically closed but assumed that the valuation is discrete. He wrote: Nach Abschlu dieses Briefes bemerke ich, da ich die Prozesse, die zu jeder moglichen diskreten Fortsetzung fuhren, vollstandig ubersehen kann. Au er den in vorstehenden Uberlegungen explizit oder implizit enthaltenen kommt noch ein wesentlich neuer Typus hinzu, bei dem der Restklassenkorper der Fortsetzung unendliche algebraische Erweiterung von K ist. Falls Sie diese vollstandige Ubersicht interessiert, so schreibe ich Ihnen gerne daruber. After writing this letter I discover that I am able to completely describe all processes which lead to all possible discrete prolongations. Besides of those which are contained explicitly oder implicitly in the above considerations, there appears a new type, in which the residue eld is an in nite algebraic extension of K . If you are interested in this complete description then I will be glad to write you about this. Unfortunately this topic appears neither in later letters of F. K. Schmidt nor in his publications. So we will never know presicsely what F. K. Schmidt had discovered. But from those remarks it does not seem impossible that he had already the full solution which later was given by Mac Lane in his paper 1936f]. That paper of Mac Lane appeared two years after Ostrowski's 1934]. Mac Lane cites Ostrowski but, he says, his description is di erent. However, although formally it is indeed di erent, it is not di cult to subsume Mac Lane's results under those of Ostrowski. Let us brie y report about 1936f]: Let us write the valuations additively. The given valuation on K is denoted by v. Suppose v is extended to a valuation w of K (x). Then w can be approximated as follows: LetP = w(x). First consider the valuation v1 on K x] de ned for polynomials f (x) = 0 i n ai xi by v1 (f (x)) = 0minn (v(ai ) + i ) : i This is the valuation with x as an \invariable element" in the terminology of Ostrowski, as mentioned earlier already. Due to the non-archimedean triangle inequality we have v1 (f (x)) w(f (x)) (33) for every polynomial f (x) 2 K x]. This v1 is called the rst approximation to w. We have either v1 = w, or there is a polynomial '(x) such that v1 ('(x)) < w('(x)): 58 Peter Roquette Suppose that '(x) is monic of smallest degree with this property; in particular '(x) is irreducible. Put 1 = w('(x)). Every polynomial f (x) 2 K x] admits a unique expansion of the form X f (x) = ai (x)'(x)i 0 in with ai (x) 2 K x] and deg ai (x) < deg '(x). Then put v2 (f (x)) = 0minn (v1 (ai(x)) + i 1 ) : i This is a valuation of K x], called the second approximation of w. Now we have v1 (f (x)) v2 (f (x)) w(f (x)) : (34) If v2 6= w then we can repeat this process, thus arriving at a sequence v1 v2 v3 w of valuations of K x]. Then we have the following alternatives. Either vk = w for some k; in this case w is called a k-fold augmented valuation. 108 Or we have that v1 = limk!1 vk is a valuation of K x]. Moreover, if the original valuation v of K is discrete then Mac Lane shows that v1 = w. This leads to a constructive description of all possible valuations of K (x) if the original valuation is discrete; Mac Lane carefully describes the properties of those poynomials '(x) which appear in the successive approximations; he calls them \key polynomials". If the valuation w is not of augmented type then the value group of w is commensurable with the value group of v, and the residue eld of K (x) with respect to w is the union of the increasing sequence of the residue elds of the approximants vk , each of which is a nite extension of K . We see that F. K. Schmidt's observations, as mentioned in his 1930 letter to Hasse, are quite in accordance with these results of Mac Lane. The motivation of Mac Lane for this investigation was to obtain a systematic explanation of various irreducibility criteria for polynomials, like the Eisenstein criterium; such irreducibility criteria had been given e.g., by Kurschak 1923e], Ore 1927e], Rella 1927d]. For details we refer to Mac Lane's paper 1938] which, as Kaplansky 1979a] points out, stands as de nitive to this day. We also remark that the said limit construction of valuations w of K (x) is of the same type as Zariski has constructed in his investigations on algebraic surfaces, in particular by obtaining valuations of a two-dimensional function eld by succesive blow-ups. But this belongs to the subject of the second part of our project, so we will not discuss it here. 5.3.3 The general valuation problem. Let us return to Ostrowski's paper 1934], Part III. There, Ostrowski puts the \general valuation problem" as follows: Given a valued eld K and an arbitrary extension eld L of K , one should give a method to construct all possible extensions of the valuation from K to L. The solution is, he says, as follows: Let T = (ti )i2I be a basis of transcendency of LjK . Then LjK (T ) is algebraic. The extensions of a valuation to an algebraic eld extension are decribed in Part II. Hence it su ces to deal with the purely transcendental extension K (T ). Now, he says, the index set I may be assumed to be well ordered, and so K (T ) is obtained from K by successively adjoining one transcendental element ti to the eld constructed in the foregoing steps. Hence it su ces to deal with just one transcendental element, and this case is covered by Ostrowski's theory of pseudoCauchy sequences. 108 Mac Lane 1936f] speaks of \values" instead of \valuations". History of Valuation Theory, Part I 59 F. K. Schmidt in his Zentralblatt -review says this solution is purely genetic (rein genetisch ). And he compares it with the description which Hasse and himself had given of the complete discrete valued elds. This is of quite di erent nature (see section 4.3.1). He says that these results are not contained in Ostrowski's paper, which aims primarily at non-discrete valuations. Of course F. K. Schmidt is correct. It is clear that both view points, the structural one of Hasse-F. K. Schmidt and the \genetic" one by Ostrowski are of interest, and that none supersedes the other. The question is why F. K. Schmidt found it necessary to stress this self-evident point in his review ? There is a letter of F. K. Schmidt to Hasse dated Sep 9, 1933. F. K. Schmidt had been in Altdorf at the Swiss Mathematical Congress and had met Ostrowski there. The latter had informed him that just now he is reading the galley proofs of a long paper on valued elds in which he (Ostrowski) gives a precise description how to construct those elds. (Obviously these were the galley proofs for Ostrowski's paper 1934].) F. K. Schmidt reports to Hasse that upon questioning it seems that Ostrowski \had nothing essential of our things" (\ : : : da er von unseren entscheidenden Sachen nichts hat "). And he says that most probably there is no overlap between the paper of Ostrowski and their joint paper (which was expected to appear in the next days). Nevertheless F. K. Schmidt seems to have been somewhat worried because he did not nd out what precisely was contained in Ostrowski's paper. For Ostrowski in their conversation had claimed not to remember details. With this background we may perhaps understand why F. K. Schmidt had included the above mentioned sentence in his review of Ostrowski's paper. It sounds to me like a sigh of relief, because his long paper 1933c] had not been superseded by that of Ostrowski. Of course he did not yet know that 1933c] contained an error, nor that this paper was to be superseded within two years by Teichmuller and Witt. 5.3.4 The Riemann surface of a eld. Ostrowski starts the last x12 of his paper with the de nition of what today is called place of a eld. (His terminology is \Restisomorphie "). 109 He reports that to every such place there belongs a \general valuation" as de ned by Krull. He de nes a \simple" place to be one whose value group has archimedean ordering and shows that to such a \simple" place there belongs a valuation in his sense, i.e., the value group is a subgroup of the real numbers. Thus the simple places correspond to the \primes" as de ned earlier. If K jk is an algebraic function eld of several variables over an algebraically closed eld then the set of all places K ! k is called by Ostrowski the \absolute Riemann surface" of K over k. 110 Ostrowski shows that every place of this absolute Riemann surface can be obtained as a composition of simple places, i.e., of primes. If the number of these primes equals the degree of transcendency of K jk then the point is called \Puiseux point". At such a point every element in K admits a Puiseux expansion { provided K jk is of characteristic 0. We see that this last x12 is of di erent avor than the foregoing sections. The reader gets the impression that x12 had been inserted only after Ostrowski had seen the famous paper of Krull 1932g] introducing general valuation theory. We observe that Krull's paper 1932g] had appeared in 1932, and in the same year Ostrowski 109 Today we would say \Resthomomorphie ". But Ostrowski still uses the older terminology where the notion of \isomorphy" is used when we would say \homomorphy". In this terminology one distinguishes between \holoedric isomorphy" (meaning \isomorphy" in today's terminology) and \meroedric isomorphy" (which means \homomorphy"). 110 Today it is also called \Zariski space" or \Zariski-Riemann manifold". 60 Peter Roquette submitted his article (which appeared in 1934). Ostrowski strongly recommends to \read the rich article of Krull" (\ : : : die Lekture der gehaltreichen Abhandlung von Herrn Krull aufs angelegentlichste empfohlen : : : "). 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