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

More documents

Recommendations

Info

84 CHAPTER 12. DIRICHLET’S UNIT THEOREM j = 2. Set d3 = · · · = dr+s = 1. Then d1d2 = A and log(1) = 0, so r+s wi log(di) = |w1 log(d1) + w2 log(d2)| i=1 = |w1 log(d1) + w2 log(A/d1)| = |(w1 − w2)log(d1) + w2 log(A)| Since w1 = w2, we have |(w1 − w2)log(d1) + w2 log(A)| → ∞ as d1 → ∞. 12.3 Some Examples of Units in Number Fields The classical Pell’s equation is, given square-free d > 0, to find all positive integer solutions (x, y) to the equation x 2 − dy 2 = 1. Note that if x + y √ d ∈ Q( √ d), then Norm(x + y √ d) = (x + y √ d)(x − y √ d) = x 2 − dy 2 . The solutions to Pell’s equation thus form a finite-index subgroup of the group of units in the ring of integers of Q( √ d). Dirichlet’s unit theorem implies that for any d the solutions to Pell’s equation form an infinite cyclic group, a fact that takes substantial work to prove using only elementary number theory (for example, using continued fractions). We first solve the Pell equation x 2 −5y 2 = 1 by finding the units of a field using Magma (we will likely discuss algorithms for computing unit groups later in the course...). > R := PolynomialRing(RationalField()); > K := NumberField(x^2-5); > G, phi := UnitGroup(K); > G; Abelian Group isomorphic to Z/2 + Z Defined on 2 generators Relations: 2*G.1 = 0 > K!phi(G.1); -1 > u := K!phi(G.2); u; 1/2*(a + 1) > u^2; 1/2*(a + 3) > u^3; a + 2 > Norm(u); -1 > Norm(u^3);
12.3. SOME EXAMPLES OF UNITS IN NUMBER FIELDS 85 -1 > Norm(u^6); 1 > fund := u^6; > fund; 4*a + 9 > 9^2 - 5*4^2; 1 > fund^2; 72*a + 161 > fund^3; 1292*a + 2889 > fund^4; 23184*a + 51841 > fund^5; 416020*a + 930249 I think in practice for solving Pell’s equation it’s best to use the ideas in the paper [Len02]. A review of this paper says: “This wonderful article begins with history and some elementary facts and proceeds to greater and greater depth about the existence of solutions to Pell equations and then later the algorithmic issues of finding those solutions. The cattle problem is discussed, as are modern smooth number methods for solving Pell equations and the algorithmic issues of representing very large solutions in a reasonable way.” You can get the paper freely online from the Notices web page. The simplest solutions to Pell’s equation can be huge, even when d is quite small. Read Lenstra’s paper for some awesome examples from antiquity. K := NumberField(x^2-NextPrime(10^7)); > G, phi := UnitGroup(K); > K!phi(G.2); 1635802598803463282255922381210946254991426776931429155067472530\ 003400641003657678728904388162492712664239981750303094365756\ 106316392723776016806037958837914778176119741840754457028237\ 899759459100428895693238165048098039*a + 517286692885814967470170672368346798303629034373575202975075\ 605058714958080893991274427903448098643836512878351227856269\ 086856679078304979321047765031073345259902622712059164969008\ 6336036036403311756634562204182936222240930 The Magma Signature command returns the number of real and complex conjugate embeddings of K into C. The command UnitGroup, which we used above, returns the unit group UK as an abstract abelian group and a homomorphism UK → OK. Note that we have to bang (!) into K to get the units as elements of K.
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 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: 12.2. FINISHING THE PROOF OF DIRICH
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 130 and 131: 130 CHAPTER 17. ADIC NUMBERS: THE F
Page 132 and 133: 132 CHAPTER 17. ADIC NUMBERS: THE F
Page 134 and 135:
134 CHAPTER 17. ADIC NUMBERS: THE F
Page 136 and 137:
136 CHAPTER 17. ADIC NUMBERS: THE F
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?