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Nonarchimedean Local Fields Nonarchimedean Local Fields Patrick Allen Contents Introduction 4 Local fields 4 The connection with number theory6 Global fields 8 This course 9 Part 1. Nonarchimedean fields 10 1. Valuations 11 2. Valued fields and the valuation subring 13 3. Completions 16 4. Hensel's Lemma 19 5. Complete discretely valued fields 22 6. Extensions of complete valued fields 28 7. Ramification: first properties 33 8. Units 37 9. Norm groups 41 Part 2. Local class field theory 44 10. The main theorems 45 11. Formal group laws 46 12. Lubin{Tate formal groups laws 53 13. Lubin{Tate extensions 57 14. The Artin map 61 15. Coleman's norm operator 64 16. Base change 68 17. Ramification groups 69 18. The Local Kronecker{Weber Theorem 74 Part 3. Further topics? 76 Appendix A. Inverse Limits 77 Appendix B. Integral extensions, norm and trace 82 Integral extensions 82 Norm and trace 83 Bibliography 87 3 Introduction In this course we investigate the structure and Galois theory of nonarchimedean local fields. We assume the reader has had a course in graduate level algebra. Before beginning the course proper, we use this introduction to introduce general local fields, state their classification, and discuss the connections with number theory. Some of the things discussed in this introduction assume prerequisites beyond graduate level algebra. But these will not be necessary in understanding the material later, and are only meant to give the reader an initial idea of the bigger picture. Local fields A topological field is a field K such that addition, negation, multiplication, and inversion K × K ! KK ! KK × K ! KK× ! K× (x; y) 7! x + y x 7! −x (x; y) 7! xy x 7! x−1 are all continuous (K × K is given the product topology, and K× = K r f0g the subspace topology). Note that both negation and inversion are homeomorphisms as they're continuous involutions. Similarly, for any x 2 K×, the map y 7! xy is a homeomorphism with inverse y 7! x−1y. Recall that a topological space X is called discrete if every subset of X is open, and is called locally compact if every element of X is contained in a compact neighbourhood. Definition 0.1. A local field is a nondiscrete, locally compact topological field. Example 0.2. The most well known examples of local fields are R and C. Example 0.3. Fix a prime p, and define a function vp : Z ! Z [ f1g by vp(n) = k, for n 6= 0, if k k+1 a p j n but p - n, and setting vp(0) = 1. We can then extend vp to Q by setting vp( b ) = vp(a)−vp(b). This is called the p-adic valuation on Q, and we define the p-adic absolute value |·|p : Q ! R≥0 by −vp(x) jxjp = p . One checks that this is an absolute value (i.e. a multiplicative norm) on Q, hence defines a metric on Q. The completion of Q with respect to this metric is a local field, called the field of p-adic numbers (or just the p-adics), and is denoted by Qp. P n Example 0.4. Let Fp be the finite field with p elements, and let Fp((T )) := f n−∞ anT j an 2 g be the field of Laurent series over . Define |·| : ((T )) ! by jfj = p−n0 if f = P a T n Fp Fp Fp R≥0 n≥n0 n with an0 6= 0. One checks that |·| is an absolute value on Fp((T )), and that the induced topology on Fp((T )) is nondiscrete and locally compact. Example 0.5. Let K be a local field, and let L be a finite field extension of K. Giving L the topology of a finite dimensional K-vector space, L is also a local field. Note that in each of Examples 0.2 to 0.4, the topology is induced from an absolute value. This is no accident, as we'll see in the proof sketch of the following classification theorem, which shows the above examples exhaust all possibilities. Theorem 0.6. Let K be a local field. If K has characteristic 0, then K is either isomorphic to R, to C, or to a finite extension of Qp for some prime p. If K has characteristic p, then K is isomorphic to a finite extension of Fp((T )). 4 LOCAL FIELDS 5 We will not use this result in the remainder of the course, so we only give an outline of the proof. The interested reader can consult [Wei67, Chapter 1] for the details. Sketch. Since (K; +) is a locally compact abelian group, it admits a Haar measure: there is a measure µ on K such that µ(x + A) = µ(A) for any Borel subset A of K and any x 2 K. Moreover, 0 it is unique up to scalar: if µ is another choice of Haar measure, then there is α 2 R>0 such that µ0 = αµ. Fix a choice of Haar measure µ. Since multiplication by any x 2 K× is an automorphism of (K; +), precomposing µ with multiplication by x yields another Haar measure µ0. By uniqueness, there 0 is a positive real α(x) such that µ = α(x)µ. Setting α(0) = 0, we have a function α : K ! R≥0 that one proves is an absolute value 1 defining the topology on K, and moreover that K is complete with respect to α. Now assume that K has characteristic p. Since the topology on K is nondiscrete, we can find T 2 K such that 0 < α(T ) < 1. The field Fp(T ) is contained in K, and since K is complete it contains the completion of Fp(T ) with respect to αjFp(T ). One then proves that this is Fp((T )). If K has characteristic 0, then K contains Q. Since K is complete, it contains the completion of Q with respect to αjQ. Applying a theorem of Ostrowski, Theorem 0.14 below, one deduces that the restriction of α to Q is equivalent either to standard absolute value on Q, or to thep-adic absolute value for some prime p. Hence, K contains either R or Qp for some prime p. Finally, one concludes by showing that if L is a subfield of K such that the induced topology on L makes L a local field, then K is a finite extension of L. Distinguishing fields according to their characteristic is certainly a natural thing to do. However with local fields, it turns out that Qp and Fp((T )) actually behave somewhat similarly and are drastically different from R and C. This distinction is formalized in the following definition. Definition 0.7. Let K be a field and let |·| be an absolute value on K. We say |·| is archimedean if the sequence (jnj)n≥1 is unbounded, and nonarchimedean if it is bounded. If K is complete with respect to |·|, then we say K is archimedean, resp. nonarchimedean, if |·| archimedean, resp. nonarchimedean. Example 0.8. R and C are archimedean local fields, while Qp and Fp((T )) are nonarchimedean local fields. In this course, we study the nonarchimedean local fields. We'll see that their topology has a much more algebraic flavour than that of R of C. One dramatic difference is the following. If K is a nonarchimedean local field, complete for a nonarchimedean absolute value |·|, then jK×j is a discrete subgroup of R>0 (see Examples 0.3 and 0.4). This implies that open balls are also closed, so the topology on K has a base of clopen sets. In particular, this implies that K is totally disconnected. Another one of the most important differences is: Proposition 0.9. Let |·| be a nonarchimedean absolute value on a field K. For any x; y 2 K (0.10) jx + yj ≤ maxfjxj; jyjg: The inequality (0.10) is known as the ultrametric inequality. Before proving Proposition 0.9, we prove an easy lemma. Lemma 0.11. An absolute value |·| on a field K is nonarchimedean if and only if jnj ≤ 1 for all n 2 N. Proof. The \if" is obvious. Conversely, if there is a natural number m such that jmj > 1, then k (jm j)k≥1, and hence also (jnj)n≥1, is unbounded. 1 If K =∼ C, then α is the square of the usual absolute value |·| on C, hence isn't technically an absolute value as |·|2 doesn't satisfy the triangle inequality. However, it does satisfy jx + yj2 ≤ 2(jxj2 + jyj2), which is enough to do all of analysis (e.g. defines the same topology and the same notions of convergence and Cauchy sequences). THE CONNECTION WITH NUMBER THEORY 6 Proof of Proposition 0.9. Without loss of generality, we can assume jxj ≥ jyj. Then for any n−k k n n n ≥ 1 and any 0 ≤ k ≤ n, jxj jyj ≤ jxj and j k j ≤ 1 by Lemma 0.11. Thus, for any n ≥ 1, n n n X n n−k k n jx + yj = j(x + y) j ≤ jxj jyj ≤ (n + 1)jxj : k k=0 Taking nth roots, jx + yj ≤ (n + 1)1=njxj for all n ≥ 1. The result follows from taking the limit n ! 1. One consequence of this is that one can \re-centre" open balls. Precisely, let |·| be a nonarchimedean absolute value on a field K, and let B(x; r) ⊂ K be the open ball in K of radius r > 0 centred at x 2 K.
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