A Brief Introduction to Classical and Adelic Algebraic ... - William Stein

More documents

Recommendations

Info

170 CHAPTER 21. IDELES AND IDEALS This works because if w ∈ W, then either |wv| v = 1 for all v ∈ S, in which case 1 < c(w) < 2c, so w ∈ I 1 K , or |wv0 | v0 < 1 for some v0 ∈ S, in which case c(w) = v∈S |wv| v 1 · |wv0 | · · · < 2C · · · · < 1, 2C so again w ∈ I 1 K . We next show that the IK- and AK-topologies on I 1 K are the same. If x ∈ I1 K , we must show that every AK-neighborhood of x contains an AK-neighborhood and vice-versa. Let W ⊂ I 1 K be an AK-neighborhood of x. Then it contains an AK-neighborhood of the type |wv − xv| v < ε v ∈ S (21.1.2) |wv| v ≤ 1 v ∈ S (21.1.3) where S is a finite set of valuations v. This contains the IK-neighborhood in which ≤ in (21.1.2) is replaced by =. Next let H ⊂ I 1 K be an IK-neighborhood. Then it contains an IK-neighborhood of the form |wv − xv| v < ε v ∈ S (21.1.4) |wv| v = 1 v ∈ S, (21.1.5) where the finite set S contains at least all archimedean valuations v and all valuations v with |xv| v = 1. Since |xv| v = 1, we may also suppose that ε is so small that (21.1.4) implies |wv| v < 2. v Then the intersection of (21.1.4) with I 1 K is the same as that of (21.1.2) with I1 K , i.e., (21.1.4) defines an AK-neighborhood. By the product formula we have that K∗ ⊂ I1 K . The following result is of vital importance in class field theory. Theorem 21.1.12. The quotient I 1 K /K∗ with the quotient topology is compact. Proof. After the preceeding lemma, it is enough to find an AK-compact set W ⊂ AK such that the map W ∩ I 1 K → I 1 K/K ∗ is surjective. We take for W the set of w = {wv}v with |wv| v ≤ |xv| v , where x = {xv}v is any idele of content greater than the C of Lemma 20.4.1.
21.2. IDEALS AND DIVISORS 171 Let y = {yv}v ∈ I1 K . Then the content of x/y equals the content of x, so by Lemma 20.4.1 there is an a ∈ K∗ such that xv |a| v ≤ all v. Then ay ∈ W, as required. yv v Remark 21.1.13. The quotient I 1 K /K∗ is totally disconnected in the function field case. For the structure of its connected component in the number field case, see papers of Artin and Weil in the “Proceedings of the Tokyo Symposium on Algebraic Number Theory, 1955” (Science Council of Japan) or [AT90]. The determination of the character group of IK/K ∗ is global class field theory. 21.2 Ideals and Divisors Suppose that K is a finite extension of Q. Let FK be the the free abelian group on a set of symbols in bijection with the non-archimedean valuation v of K. Thus an element of FK is a formal linear combination nv · v v non arch. where nv ∈ Z and all but finitely many nv are 0. Lemma 21.2.1. There is a natural bijection between FK and the group of nonzero fractional ideals of OK. The correspondence is induced by where v is a non-archimedean valuation. v ↦→ ℘v = {x ∈ OK : v(x) < 1}, Endow FK with the discrete topology. Then there is a natural continuous map π : IK → FK given by x = {xv}v ↦→ ordv(xv) · v. This map is continuous since the inverse image of a valuation v (a point) is the product K ∗ w × O ∗ w, π −1 (v) = πO ∗ v × v w archimedean w=v non-arch. which is an open set in the restricted product topology on IK. Moreover, the image of K ∗ in FK is the group of nonzero principal fractional ideals. Recall that the class group CK of the number field K is by definition the quotient of FK by the image of K ∗ . Theorem 21.2.2. The class group CK of a number field K is finite.
Page 1:
A Brief Introduction to Classical a
Page 4 and 5:
4 CONTENTS 8 Factoring Primes 49 8.
Page 6 and 7:
6 CONTENTS
Page 8 and 9:
8 CHAPTER 1. PREFACE Acknowledgemen
Page 10 and 11:
10 CHAPTER 2. INTRODUCTION 2.2 What
Page 12 and 13:
12 CHAPTER 2. INTRODUCTION (e) Deep
Page 15 and 16:
Chapter 3 Finitely generated abelia
Page 17 and 18:
1. By permuting rows and columns, m
Page 19 and 20:
Chapter 4 Commutative Algebra We wi
Page 21 and 22:
4.1. NOETHERIAN RINGS AND MODULES 2
Page 23 and 24:
4.1. NOETHERIAN RINGS AND MODULES 2
Page 25 and 26:
Chapter 5 Rings of Algebraic Intege
Page 27 and 28:
5.2. NORMS AND TRACES 27 because Z
Page 29 and 30:
5.2. NORMS AND TRACES 29 elements o
Page 31 and 32:
Chapter 6 Unique Factorization of I
Page 33 and 34:
6.1. DEDEKIND DOMAINS 33 of a gener
Page 35 and 36:
6.1. DEDEKIND DOMAINS 35 Theorem 6.
Page 37 and 38:
Chapter 7 Computing 7.1 Algorithms
Page 39 and 40:
7.2. USING MAGMA 39 > S, P, Q := Sm
Page 41 and 42:
7.2. USING MAGMA 41 r4 + r1 > Minim
Page 43 and 44:
7.2. USING MAGMA 43 [ 1, a, 1/3*(a^
Page 45 and 46:
7.2. USING MAGMA 45 7.2.4 Ideals >
Page 47 and 48:
7.2. USING MAGMA 47 > OK := Maximal
Page 49 and 50:
Chapter 8 Factoring Primes First we
Page 51 and 52:
8.1. FACTORING PRIMES 51 [3, 1, 0,
Page 53 and 54:
8.1. FACTORING PRIMES 53 Theorem 8.
Page 55 and 56:
8.1. FACTORING PRIMES 55 Sketch of
Page 57 and 58:
Chapter 9 Chinese Remainder Theorem
Page 59 and 60:
9.1. THE CHINESE REMAINDER THEOREM
Page 61 and 62:
9.1. THE CHINESE REMAINDER THEOREM
Page 63 and 64:
Chapter 10 Discrimannts, Norms, and
Page 65 and 66:
10.2. DISCRIMINANTS 65 Lemma 10.2.1
Page 67 and 68:
10.4. FINITENESS OF THE CLASS GROUP
Page 69 and 70:
Page 71 and 72:
Page 73 and 74:
Chapter 11 Computing Class Groups I
Page 75 and 76:
11.1. REMARKS ON COMPUTING THE CLAS
Page 77 and 78:
Chapter 12 Dirichlet’s Unit Theor
Page 79 and 80:
12.1. THE GROUP OF UNITS 79 Lemma 1
Page 81 and 82:
12.2. FINISHING THE PROOF OF DIRICH
Page 83 and 84:
12.2. FINISHING THE PROOF OF DIRICH
Page 85 and 86:
12.3. SOME EXAMPLES OF UNITS IN NUM
Page 87 and 88:
12.3. SOME EXAMPLES OF UNITS IN NUM
Page 89 and 90:
12.4. PREVIEW 89 > time G,phi := Un
Page 91 and 92:
Chapter 13 Decomposition and Inerti
Page 93 and 94:
13.2. DECOMPOSITION OF PRIMES 93 If
Page 95 and 96:
13.2. DECOMPOSITION OF PRIMES 95 Th
Page 97 and 98:
Chapter 14 Decomposition Groups and
Page 99 and 100:
14.1. THE DECOMPOSITION GROUP 99 Th
Page 101 and 102:
14.2. FROBENIUS ELEMENTS 101 Figure
Page 103 and 104:
14.3. GALOIS REPRESENTATIONS AND A
Page 105:
Part II Adelic Viewpoint 105
Page 108 and 109:
108 CHAPTER 15. VALUATIONS Note tha
Page 110 and 111:
110 CHAPTER 15. VALUATIONS To say t
Page 112 and 113:
112 CHAPTER 15. VALUATIONS Proof. F
Page 114 and 115:
114 CHAPTER 15. VALUATIONS 1 so |u|
Page 116 and 117:
116 CHAPTER 15. VALUATIONS of a val
Page 118 and 119:
118 CHAPTER 16. TOPOLOGY AND COMPLE
Page 120 and 121: 120 CHAPTER 16. TOPOLOGY AND COMPLE
Page 130 and 131: 130 CHAPTER 17. ADIC NUMBERS: THE F
Page 138 and 139: 138 CHAPTER 18. NORMED SPACES AND T
Page 146 and 147: 146CHAPTER 19. EXTENSIONS AND NORMA
Page 154 and 155: 154 CHAPTER 20. GLOBAL FIELDS AND A
Page 168 and 169: 168 CHAPTER 21. IDELES AND IDEALS E
Page 172 and 173: 172 CHAPTER 21. IDELES AND IDEALS P
Page 174 and 175: 174 CHAPTER 22. EXERCISES 6. Find t
Page 176 and 177: 176 CHAPTER 22. EXERCISES 22. (*) S
Page 178 and 179: 178 CHAPTER 22. EXERCISES (a) The p
Page 180 and 181: 180 CHAPTER 22. EXERCISES 60. Find
Page 182 and 183: 182 CHAPTER 22. EXERCISES
Page 184 and 185: 184 BIBLIOGRAPHY [Fre94] G. Frey (e
Page 186 and 187: 186 INDEX complex n-dimensional rep
Page 188 and 189: 188 INDEX lies over, 73 local compa
Page 190: 190 INDEX valuations such that |a|
show all

A Brief Introduction to Classical and Adelic Algebraic ... - William Stein

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?