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

More documents

Recommendations

Info

28 CHAPTER 5. RINGS OF ALGEBRAIC INTEGERS Note that if f ∈ Q[x] is the characteristic polynomial of ℓa, then the constant term of f is (−1) deg(f) Det(ℓa), and the coefficient of x deg(f)−1 is − tr(ℓa). Proposition 5.2.2. Let a ∈ L and let σ1, . . .,σd, where d = [L : K], be the distinct field embeddings L ↩→ Q that fix every element of K. Then Norm L/K(a) = d σi(a) and trL/K(a) = i=1 d σi(a). Proof. We prove the proposition by computing the characteristic polynomial F of a. Let f ∈ K[x] be the minimal polynomial of a over K, and note that f has distinct roots (since it is the polynomial in K[x] of least degree that is satisfied by a). Since f is irreducible, [K(a) : K] = deg(f), and a satisfies a polynomial if and only if ℓa does, the characteristic polynomial of ℓa acting on K(a) is f. Let b1, . . .,bn be a basis for L over K(a) and note that 1, . . .,a m is a basis for K(a)/K, where m = deg(f) −1. Then a i bj is a basis for L over K, and left multiplication by a acts the same way on the span of bj, abj, . . .,a m bj as on the span of bk, abk, . . .,a m bk, for any pair j, k ≤ n. Thus the matrix of ℓa on L is a block direct sum of copies of the matrix of ℓa acting on K(a), so the characteristic polynomial of ℓa on L is f [L:K(a)] . The proposition follows because the roots of f [L:K(a)] are exactly the images σi(a), with multiplicity [L : K(a)] (since each embedding of K(a) into Q extends in exactly [L : K(a)] ways to L by Exercise 9). The following corollary asserts that the norm and trace behave well in towers. Corollary 5.2.3. Suppose K ⊂ L ⊂ M is a tower of number fields, and let a ∈ M. Then Norm M/K(a) = Norm L/K(Norm M/L(a)) and tr M/K(a) = tr L/K(tr M/L(a)). Proof. For the first equation, both sides are the product of σi(a), where σi runs through the embeddings of M into K. To see this, suppose σ : L → Q fixes K. If σ ′ is an extension of σ to M, and τ1, . . .,τd are the embeddings of M into Q that fix L, then τ1σ ′ , . . .,τdσ ′ are exactly the extensions of σ to M. For the second statement, both sides are the sum of the σi(a). The norm and trace down to Q of an algebraic integer a is an element of Z, because the minimal polynomial of a has integer coefficients, and the characteristic polynomial of a is a power of the minimal polynomial, as we saw in the proof of Proposition 5.2.2. Proposition 5.2.4. Let K be a number field. The ring of integers OK is a lattice in K, i.e., QOK = K and OK is an abelian group of rank [K : Q]. Proof. We saw in Lemma 5.1.8 that QOK = K. Thus there exists a basis a1, . . .,an for K, where each ai is in OK. Suppose that as x = ciai ∈ OK varies over all i=1
5.2. NORMS AND TRACES 29 elements of OK the denominators of the coefficients ci are arbitrarily large. Then subtracting off integer multiples of the ai, we see that as x = ciai ∈ OK varies over elements of OK with ci between 0 and 1, the denominators of the ci are also arbitrarily large. This implies that there are infinitely many elements of OK in the bounded subset S = {c1a1 + · · · + cnan : ci ∈ Q, 0 ≤ ci ≤ 1} ⊂ K. Thus for any ε > 0, there are elements a, b ∈ OK such that the coefficients of a − b are all less than ε (otherwise the elements of OK would all be a “distance” of least ε from each other, so only finitely many of them would fit in S). As mentioned above, the norms of elements of OK are integers. Since the norm of an element is the determinant of left multiplication by that element, the norm is a homogenous polynomial of degree n in the indeterminate coefficients ci. If the ci get arbitrarily small for elements of OK, then the values of the norm polynomial get arbitrarily small, which would imply that there are elements of OK with positive norm too small to be in Z, a contradiction. So the set S contains only finitely many elements of OK. Thus the denominators of the ci are bounded, so for some d, we have that OK has finite index in A = 1 dZa1 + · · · + 1 dZan. Since A is isomorphic to Z n , it follows from the structure theorem for finitely generated abelian groups that OK is isomorphic as a Z-module to Z n , as claimed. Corollary 5.2.5. The ring of integers OK of a number field is Noetherian. Proof. By Proposition 5.2.4, the ring OK is finitely generated as a module over Z, so it is certainly finitely generated as a ring over Z. By the Hilbert Basis Theorem, OK is Noetherian.
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: 5.2. NORMS AND TRACES 27 because Z
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 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:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
130 CHAPTER 17. ADIC NUMBERS: THE F
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
138 CHAPTER 18. NORMED SPACES AND T
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
146CHAPTER 19. EXTENSIONS AND NORMA
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
154 CHAPTER 20. GLOBAL FIELDS AND A
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
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

Create successful ePaper yourself

Delete template?

Save as template?