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

More documents

Recommendations

Info

42 CHAPTER 7. COMPUTING Maximal Equation Order with defining polynomial x^2 + [0, -1, 0] over its ground order Sometimes the ring of integers of Q(a) isn’t just Z[a]. First a simple example, then a more complicated one: > K := NumberField(2*x^2-3); // doesn’t have to be monic > 2*a^2 - 3; 0 > K; Number Field with defining polynomial x^2 - 3/2 over the Rational Field > O := RingOfIntegers(K); > O; Maximal Order of Equation Order with defining polynomial 2*x^2 - 3 over ZZ > Basis(O); [ O.1, O.2 ] > [K!x : x in Basis(O)]; [ 1, 2*a // this is Sqrt(3) ] Here’s are some more examples: > procedure ints(f) // (procedures don’t return anything; functions do) K := NumberField(f); O := MaximalOrder(K); print [K!z : z in Basis(O)]; end procedure; > ints(x^2-5); [ 1, 1/2*(a + 1) ] > ints(x^2+5); [ 1, a ] > ints(x^3-17);
7.2. USING MAGMA 43 [ 1, a, 1/3*(a^2 + 2*a + 1) ] > ints(CyclotomicPolynomial(7)); [ 1, a, a^2, a^3, a^4, a^5 ] > ints(x^5+&+[Random(10)*xî : i in [0..4]]); // RANDOM [ 1, a, a^2, a^3, a^4 ] > ints(x^5+&+[Random(10)*xî : i in [0..4]]); // RANDOM [ 1, a, a^2, 1/2*(a^3 + a), 1/16*(a^4 + 7*a^3 + 11*a^2 + 7*a + 14) ] Lets find out how high of a degree Magma can easily deal with. > d := 10; time ints(x^10+&+[Random(10)*xî : i in [0..d-1]]); [ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9 ] Time: 0.030 > d := 15; time ints(x^10+&+[Random(10)*xî : i in [0..d-1]]); [ 1, 7*a, 7*a^2 + 4*a, 7*a^3 + 4*a^2 + 4*a, 7*a^4 + 4*a^3 + 4*a^2 + a,
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: 7.2. USING MAGMA 41 r4 + r1 > Minim
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

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

Delete template?

Save as template?