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

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70CHAPTER 10. DISCRIMANNTS, NORMS, AND FINITENESS OF THE CLASS GROUP Let S ⊂ V be any closed, bounded, convex, subset that is symmetric with respect to the origin and has positive volume. Since S is closed and bounded, M = max{f(x) : x ∈ S} exists. Suppose I is any nonzero fractional ideal of OK. Our goal is to prove there is an integral ideal aI with small norm. We will do this by finding an appropriate a ∈ I −1 . By Lemma 10.4.6, c = Vol(V/I −1 ) = 2−s |dK| Norm(I) . Let λ = 2 · c 1/n, v where v = Vol(S). Then Vol(λS) = λ n n c Vol(S) = 2 v · v = 2n · c = 2 n Vol(V/I −1 ), so by Lemma 10.4.3 there exists 0 = a ∈ I −1 ∩ λS. Since M is the largest norm of an element of S, the largest norm of an element of I −1 ∩ λS is at most λ n M, so | Norm K/Q(a)| ≤ λ n M. Since a ∈ I −1 , we have aI ⊂ OK, so aI is an integral ideal of OK that is equivalent to I, and Norm(aI) = | Norm K/Q(a)| · Norm(I) ≤ λ n M · Norm(I) n c ≤ 2 M · Norm(I) v ≤ 2 n · 2 −s |dK| · M · v −1 = 2 r+s |dK| · M · v −1 . Notice that the right hand side is independent of I. It depends only on r, s, |dK|, and our choice of S. This completes the proof of the theorem, except for the assertion that S can be chosen to give the claim at the end of the theorem, which we leave as an exercise. Corollary 10.4.7. Suppose that K = Q is a number field. Then |dK| > 1. Proof. Applying Theorem 10.4.2 to the unit ideal, we get the bound 1 ≤ s 4 n! |dK| · π nn. Thus π s n |dK| ≥ 4 n n! , and the right hand quantity is strictly bigger than 1 for any s ≤ n/2 and any n > 1 (exercise).
10.4. FINITENESS OF THE CLASS GROUP VIA GEOMETRY OF NUMBERS71 10.4.1 An Open Problem Conjecture 10.4.8. There are infinitely many number fields K such that the class group of K has order 1. For example, if we consider real quadratic fields K = Q( √ d), with d positive and square free, many class numbers are probably 1, as suggested by the Magma output below. It looks like 1’s will keep appearing infinitely often, and indeed Cohen and Lenstra conjecture that they do. Nobody has found a way to prove this yet. > for d in [2..1000] do if d eq SquareFree(d) then h := ClassNumber(NumberField(x^2-d)); if h eq 1 then printf "%o, ", d; end if; end if; end for; 2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 21, 22, 23, 29, 31, 33, 37, 38, 41, 43, 46, 47, 53, 57, 59, 61, 62, 67, 69, 71, 73, 77, 83, 86, 89, 93, 94, 97, 101, 103, 107, 109, 113, 118, 127, 129, 131, 133, 134, 137, 139, 141, 149, 151, 157, 158, 161, 163, 166, 167, 173, 177, 179, 181, 191, 193, 197, 199, 201, 206, 209, 211, 213, 214, 217, 227, 233, 237, 239, 241, 249, 251, 253, 262, 263, 269, 271, 277, 278, 281, 283, 293, 301, 302, 307, 309, 311, 313, 317, 329, 331, 334, 337, 341, 347, 349, 353, 358, 367, 373, 379, 381, 382, 383, 389, 393, 397, 398, 409, 413, 417, 419, 421, 422, 431, 433, 437, 446, 449, 453, 454, 457, 461, 463, 467, 478, 479, 487, 489, 491, 497, 501, 502, 503, 509, 517, 521, 523, 526, 537, 541, 542, 547, 553, 557, 563, 566, 569, 571, 573, 581, 587, 589, 593, 597, 599, 601, 607, 613, 614, 617, 619, 622, 631, 633, 641, 643, 647, 649, 653, 661, 662, 669, 673, 677, 681, 683, 691, 694, 701, 709, 713, 717, 718, 719, 721, 734, 737, 739, 743, 749, 751, 753, 757, 758, 766, 769, 773, 781, 787, 789, 797, 809, 811, 813, 821, 823, 827, 829, 838, 849, 853, 857, 859, 862, 863, 869, 877, 878, 881, 883, 886, 887, 889, 893, 907, 911, 913, 917, 919, 921, 926, 929, 933, 937, 941, 947, 953, 958, 967, 971, 973, 974, 977, 983, 989, 991, 997, 998, In contrast, if we look at class numbers of quadratic imaginary fields, only a few at the beginning have class number 1. > for d in [1..1000] do if d eq SquareFree(d) then h := ClassNumber(NumberField(x^2+d));
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A Brief Introduction to Classical a
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4 CONTENTS 8 Factoring Primes 49 8.
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6 CONTENTS
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8 CHAPTER 1. PREFACE Acknowledgemen
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10 CHAPTER 2. INTRODUCTION 2.2 What
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12 CHAPTER 2. INTRODUCTION (e) Deep
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Chapter 3 Finitely generated abelia
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1. By permuting rows and columns, m
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Page 57 and 58: Chapter 9 Chinese Remainder Theorem
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Page 73 and 74: Chapter 11 Computing Class Groups I
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Page 101 and 102: 14.2. FROBENIUS ELEMENTS 101 Figure
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Page 105: Part II Adelic Viewpoint 105
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120 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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