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

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44 CHAPTER 7. COMPUTING 7*a^5 + 4*a^4 + 4*a^3 + a^2 + a, 7*a^6 + 4*a^5 + 4*a^4 + a^3 + a^2 + 4*a, 7*a^7 + 4*a^6 + 4*a^5 + a^4 + a^3 + 4*a^2, 7*a^8 + 4*a^7 + 4*a^6 + a^5 + a^4 + 4*a^3 + 4*a, 7*a^9 + 4*a^8 + 4*a^7 + a^6 + a^5 + 4*a^4 + 4*a^2 + 5*a, 7*a^10 + 4*a^9 + 4*a^8 + a^7 + a^6 + 4*a^5 + 4*a^3 + 5*a^2 + 4*a, ... ] Time: 0.480 > d := 20; time ints(x^10+&+[Random(10)*x^i : i in [0..d-1]]); [ 1, 2*a, 4*a^2, 8*a^3, 8*a^4 + 2*a^2 + a, 8*a^5 + 2*a^3 + 3*a^2, ...] Time: 3.940 > d := 25; time ints(x^10+&+[Random(10)*x^i : i in [0..d-1]]); ... I stopped it after a few minutes... We can also define orders in rings of integers. > R := PolynomialRing(RationalField()); > K := NumberField(x^3-2); > O := Order([2*a]); > O; Transformation of Order over Equation Order with defining polynomial x^3 - 2 over ZZ Transformation Matrix: [1 0 0] [0 2 0] [0 0 4] > OK := MaximalOrder(K); > Index(OK,O); 8 > Discriminant(O); -6912 > Discriminant(OK); -108 > 6912/108; 64 // perfect square...
7.2. USING MAGMA 45 7.2.4 Ideals > R := PolynomialRing(RationalField()); > K := NumberField(x^3-2); > O := Order([2*a]); > O; Transformation of Order over Equation Order with defining polynomial x^3 - 2 over ZZ Transformation Matrix: [1 0 0] [0 2 0] [0 0 4] > OK := MaximalOrder(K); > Index(OK,O); 8 > Discriminant(O); -6912 > Discriminant(OK); -108 > 6912/108; 64 // perfect square... > R := PolynomialRing(RationalField()); > K := NumberField(x^2-7); > K := NumberField(x^2-5); > Discriminant(K); 20 // ????????? Yuck! > OK := MaximalOrder(K); > Discriminant(OK); 5 // better > Discriminant(NumberField(x^2-20)); 80 > I := 7*OK; > I; Principal Ideal of OK Generator: [7, 0] > J := (OK!a)*OK; // the ! computes the natural image of a in OK > J; Principal Ideal of OK Generator: [-1, 2] > I*J; Principal Ideal of OK Generator:
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: 7.2. USING MAGMA 43 [ 1, a, 1/3*(a^
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
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13.2. DECOMPOSITION OF PRIMES 95 Th
Page 97 and 98:
Chapter 14 Decomposition Groups and
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14.1. THE DECOMPOSITION GROUP 99 Th
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14.2. FROBENIUS ELEMENTS 101 Figure
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14.3. GALOIS REPRESENTATIONS AND A
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Part II Adelic Viewpoint 105
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108 CHAPTER 15. VALUATIONS Note tha
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110 CHAPTER 15. VALUATIONS To say t
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112 CHAPTER 15. VALUATIONS Proof. F
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114 CHAPTER 15. VALUATIONS 1 so |u|
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116 CHAPTER 15. VALUATIONS of a val
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118 CHAPTER 16. TOPOLOGY AND COMPLE
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130 CHAPTER 17. ADIC NUMBERS: THE F
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138 CHAPTER 18. NORMED SPACES AND T
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146CHAPTER 19. EXTENSIONS AND NORMA
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154 CHAPTER 20. GLOBAL FIELDS AND A
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168 CHAPTER 21. IDELES AND IDEALS E
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170 CHAPTER 21. IDELES AND IDEALS T
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172 CHAPTER 21. IDELES AND IDEALS P
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174 CHAPTER 22. EXERCISES 6. Find t
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176 CHAPTER 22. EXERCISES 22. (*) S
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178 CHAPTER 22. EXERCISES (a) The p
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180 CHAPTER 22. EXERCISES 60. Find
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182 CHAPTER 22. EXERCISES
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184 BIBLIOGRAPHY [Fre94] G. Frey (e
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186 INDEX complex n-dimensional rep
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188 INDEX lies over, 73 local compa
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190 INDEX valuations such that |a|
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A Brief Introduction to Classical and Adelic Algebraic ... - William Stein

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