8410Chapter1

# 8410Chapter1 - LECTURE NOTES ON VALUATION THEORY PETE L...

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Unformatted text preview: LECTURE NOTES ON VALUATION THEORY PETE L. CLARK Contents 1. Absolute values and valuations 1 1.1. Basic definitions 1 1.2. Absolute values and the Artin constant 2 1.3. Equivalence of absolute values 4 1.4. Artin-Whaples Approximation Theorem 5 1.5. Archimedean absolute values 7 1.6. Non-archimedean norms and valuations 9 1.7. R-regular valuations; valuations on Dedekind domains 11 1.8. Some Classification Theorems 12 References 16 1. Absolute values and valuations 1.1. Basic definitions. All rings are commutative with unity unless explicit mention is made otherwise. A norm , on a field k is a map | | : k → R ≥ satisfying: (V1) | x | = 0 ⇐⇒ x = 0. (V2) ∀ x,y ∈ k, | xy | = | x || y | . (V3) ∀ x,y ∈ k , | x + y | ≤ | x | + | y | . Example 1.1.0: On any field k , define | | : k → R ≥ by 0 7→ , x ∈ k \ { } 7→ 1. This is immediately seen to be an absolute value on k (Exercise!), called the trivial norm . In many respects it functions as an exception in the general theory. Example 1.1.1: The standard norm on the complex numbers: | a + bi | = √ a 2 + b 2 . The restriction of this to Q or to R will also be called “standard”. Example 1.1.2: The p-adic norm on Q : write a b = p n c d with gcd( p, cd ) = 1 and put | a b | p = p- n . We will see more examples later. In particular, to each prime ideal in a Dedekind Thanks to John Doyle and David Krumm for pointing out typos. 1 2 PETE L. CLARK domain we may associate a norm. This remark serves to guarantee both the pleni- tude of examples of norms and their link to classical algebraic number theory. Exercise 1.1: Let | | be a norm on the field k . Show that for all a,b ∈ k , || a | - | b || ≤ | a- b | . Exercise 1.2: Let R be a ring which is not the zero ring (i.e., 1 6 = 0 in R ). Let | | : R → R ≥ be a map which satisfies (V1) and (V2) (with k replaced by R ) above. a) Show that | 1 | = 1. b) Show that R is an integral domain, hence has a field of fractions k . c) Show that there is a unique extension of | | to k , the fraction field of R , satisfying (V1) and (V2). d) Suppose that moreover R satisfies (V3). Show that the extension of part b) to k satisfies (V3) and hence defines a norm on k . e) Conversely, show that every integral domain admits a mapping | | satisfying (V1), (V2), (V3). Exercise 1.3: a) Let | | be a norm on k and x ∈ k a root of unity. 1 Show | x | = 1. b) Show that for a field k , TFAE: (i) Every nonzero element of k is a root of unity. (ii) The characteristic of k is p > 0, and k/ F p is algebraic. c) If k/ F p is algebraic, show that the only norm on k is | | . Remark: In Chapter 2 we will see that the converse of Exercise 1.3c) is also true: any field which is not algebraic over a finite field admits at least one (and in fact infinitely many) nontrivial norm....
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8410Chapter1 - LECTURE NOTES ON VALUATION THEORY PETE L...

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