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

More documents

Recommendations

Info

130 CHAPTER 17. ADIC NUMBERS: THE FINITE RESIDUE FIELD CASE (an), there is N such that for n, m ≥ N, we have |an − am| < ε. In particular, we can choose N such that n, m ≥ N implies that an ≡ am (mod p). Let ϕ((an)) = aN (mod p), which is well-defined. The map ϕ is surjective because the constant sequences are in Ov. Its kernel is the set of Cauchy sequences whose elements are eventually all in p, which is exactly pv. This proves the first part of the lemma. The second part is true because any element of pv is a sequence all of whose terms are eventually in p, hence all a multiple of π (we can set to 0 a finite number of terms of the sequence without changing the equivalence class of the sequence). Assume for the rest of this section that K is complete with respect to | · |. Lemma 17.1.2. Then ring O is precisely the set of infinite sums a = ∞ j=0 aj · π j (17.1.1) where the aj run independently through some set R of representatives of O in O/p. By (17.1.1) is meant the limit of the Cauchy sequence n j=0 aj · π j as j → ∞. Proof. There is a uniquely defined a0 ∈ R such that |a − a0| < 1. Then a ′ = π −1 · (a − a0) ∈ O. Now define a1 ∈ R by |a ′ − a1| < 1. And so on. Example 17.1.3. Suppose K = Q and | · | = | · | p is the p-adic valuation, for some prime p. We can take R = {0, 1, . . .,p − 1}. The lemma asserts that ⎧ ⎨ ∞ O = Zp = anp ⎩ n ⎫ ⎬ : 0 ≤ an ≤ p − 1 ⎭ . j=0 Notice that O is uncountable since there are p choices for each p-adic “digit”. We can do arithmetic with elements of Zp, which can be thought of “backwards” as numbers in base p. For example, with p = 3 we have (1 + 2 · 3 + 3 2 + · · ·) + (2 + 2 · 3 + 3 2 + · · ·) = 3 + 4 · 3 + 2 · 3 2 + · · · not in canonical form = 0 + 2 · 3 + 3 · 3 + 2 · 3 2 + · · · still not canonical = 0 + 2 · 3 + 0 · 3 2 + · · · Basic arithmetic with the p-adics in Magma is really weird (even weirder than it was a year ago... There are presumably efficiency advantages to using the Magma formalization, and it’s supposed to be better for working with extension fields. But I can’t get it to do even the calculation below in a way that is clear.) In PARI (gp) the p-adics work as expected:
17.1. FINITE RESIDUE FIELD CASE 131 ? a = 1 + 2*3 + 3^2 + O(3^3); ? b = 2 + 2*3 + 3^2 + O(3^3); ? a+b %3 = 2*3 + O(3^3) ? sqrt(1+2*3+O(3^20)) %5 = 1 + 3 + 3^2 + 2*3^4 + 2*3^7 + 3^8 + 3^9 + 2*3^10 + 2*3^12 + 2*3^13 + 2*3^14 + 3^15 + 2*3^17 + 3^18 + 2*3^19 + O(3^20) ? 1/sqrt(1+2*3+O(3^20)) %6 = 1 + 2*3 + 2*3^2 + 2*3^7 + 2*3^10 + 2*3^11 + 2*3^12 + 2*3^13 + 2*3^14 + 3^15 + 2*3^16 + 2*3^17 + 3^18 + 3^19 + O(3^20) Theorem 17.1.4. Under the conditions of the preceding lemma, O is compact with respect to the | · |-topology. Proof. Let Vλ, for λ running through some index set Λ, be some family of open sets that cover O. We must show that there is a finite subcover. We suppose not. Let R be a set of representatives for O/p. Then O is the union of the finite number of cosets a + πO, for a ∈ R. Hence for at lest one a0 ∈ R the set a0 + πO is not covered by finitely many of the Vλ. Then similarly there is an a1 ∈ R such that a0 + a1π + π 2 O is not finitely covered. And so on. Let a = a0 + a1π + a2π 2 + · · · ∈ O. Then a ∈ Vλ0 for some λ0 ∈ Λ. Since Vλ0 is an open set, a+πJ · O ⊂ Vλ0 for some J (since those are exactly the open balls that form a basis for the topology). This is a contradiction because we constructed a so that none of the sets a + πn · O, for each n, are not covered by any finite subset of the Vλ. Definition 17.1.5 (Locally compact). A topological space X is locally compact at a point x if there is some compact subset C of X that contains a neighborhood of x. The space X is locally compact if it is locally compact at each point in X. Corollary 17.1.6. The complete local field K is locally compact. Proof. If x ∈ K, then x ∈ C = x + O, and C is a compact subset of K by Theorem 17.1.4. Also C contains the neighborhood x + πO = B(x,1) of x. Thus K is locally compact at x. Remark 17.1.7. The converse is also true. If K is locally compact with respect to a non-archimedean valuation | · |, then 1. K is complete, 2. the residue field is finite, and 3. the valuation is discrete.
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 132 and 133: 132 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 170 and 171: 170 CHAPTER 21. IDELES AND IDEALS T
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?