Sudhir R. Ghorpade - Notes on Galois Theory Sudhir R....

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Unformatted text preview: Notes on Galois Theory Sudhir R. Ghorpade Department of Mathematics, Indian Institute of Technology, Bombay 400 076 E-mail : srg@math.iitb.ac.in October 1994 Contents 1 Preamble 2 2 Field Extensions 3 3 Splitting Fields and Normal Extensions 6 4 Separable Extensions 9 5 Galois Theory 11 6 Norms and Traces 16 1 1 Preamble These notes attempt to give an introduction to some basic aspects of Field Theory and Galois Theory. Originally, the succeeding sections of these notes constituted a part of the notes prepared to supplement the lectures of the author on Galois Theory and Ramification Theory at the All India Summer School in Number Theory held at Pune in June 1991. Subsequently, the first 6 sections of the Pune Notes were separated and slightly revised to form these “Notes on Galois Theory”, which were used for pre-conference distribution to the participants of the NBHM sponsored Instructional School on Algebraic Number Theory (University of Bombay, December 1994) at the request of the organisers. A few minor revisions have taken place in the subsequent years. The main aim of these notes has always been to provide a geodesic, yet complete, presen- tation starting from the definition of field extensions and concluding with the Fundamental Theorem of Galois Theory. Some additional material on separable extensions and a section on Norms and Traces is also included, and some historical comments appear as footnotes. The prerequisite for these notes is basic knowledge of Abstract Algebra and Linear Algebra not beyond the contents of usual undergraduate courses in these subjects. No formal background in Galois Theory is assumed. While a complete proof of the Fundamental Theorem of Galois Theory is given here, we do not discuss further results such as Galois’ theorem on solvability of equations by radicals. An annotated list of references for Galois Theory appears at the end of Section 5. By way of references for the last section, viz., Norms and Traces, we recommend Van der Waerden’s “Algebra” (F. Ungar Pub. Co., 1949) and Zariski–Samuel’s “Commutative Algebra, Vol. 1” (Springer-Verlag, 1975). It appears that over the years, these notes are often used by students primarily interested in Number Theory. Thus it may be pertinent to remark at the outset that the topics discussed in these notes are very useful in the study of Algebraic Number Theory 1 . In order to derive maximum benefit from these notes, the students are advised to attempt all the Exercises and fill the missing steps, if any, in the proofs given. The author would appreciate receiving comments, suggestions and criticism regarding these notes. 1 In fact, questions concerning integers alone, can sometimes be answered only with the help of field ex- tensions and certain algebraic objects associated to them. For instance, Kummer showed that the equation X p + Y p = Z p has no integer solution for a class of odd primes p , called regular primes, which include all odd primes less than 100 except 37, 59 and 67. Even a convenient definition of regular primes, not to mention theprimes less than 100 except 37, 59 and 67....
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Sudhir R. Ghorpade - Notes on Galois Theory Sudhir R....

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