NUMBER THEORY An Introduction to Mathematics: Part A NUMBER THEORY An Introduction to Mathematics: Part A BY WILLIAM A COPPEL - Springer Library of Congress Control Number: 2005934653 PARTA ISBN-10: 0-387-29851-7 e-ISBN: 0-387-29852-5 ISBN-13: 978-0387-29851-1 PART B ISBN-10: 0-387-29853-3 e-ISBN: 0-387-29854-1 ISBN-13: 978-0387-29853-5 PVOLUME SET ISBN-10: 0-387-30019-8 e-ISBN:0-387-30529-7 ISBN-]3: 978-0387-30019-1 Printed on acid-free paper AMS Subiect Classifications: 1-xx 05820 33E05 O 2006 Springer Science+Business Media, Inc All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Strcct, Ncw York, NY 10013, USA), except for brief exccrpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, clcctronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade narncs, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights Printed in the United States of America For Jonathan, Nicholas, Philip and Stephen Contents Part A Preface I The expanding universe of numbers Sets, relations and mappings Natural numbers Integers and rational numbers Real numbers Metric spaces Complex numbers Quaternions and octonions Groups Rings and fields Vector spaces and associative algebras Inner product spaces Further remarks Selected references I1 Divisibility Greatest common divisors The Bezout identity Polynomials Euclidean domains Congruences Sums of squares Further remarks Selected references Contents Vlll I11 More on divisibility The law of quadratic reciprocity IV Quadratic fields Multiplicative functions Linear Diophantine equations Further remarks Selected references Continued fractions and their uses The continued fraction algorithm V Diophantine approximation Periodic continued fractions Quadratic Diophantine equations The modular group Non-Euclidean geometry Complements Further remarks Selected references Hadamard's determinant problem What is a determinant? Hadamard matrices The art of weighing Some matrix theory Application to Hadamard's determinant problem Designs Groups and codes Further remarks Selected references Contents VI Hensel's p-adic numbers Valued fields Equivalence Completions Non-archimedean valued fields Hensel's lemma Locally compact valued fields Further remarks Selected references Notations Axioms Index Part B VII The arithmetic of quadratic forms Quadratic spaces The Hilbert symbol The Hasse-Minkowski theorem Supplements Further remarks Selected references VIII The geometry of numbers Minkowski's lattice point theorem Lattices Proof of the lattice-point theorem, and some generalizations Voronoi cells Densest packings Mahler's compactness theorem Further remarks Selected references Contents IX The number of prime numbers X Finding the problem Chebyshev's functions Proof of the prime number theorem The Riemann hypothesis Generalizations and analogues Alternative formulations Some further problems Further remarks Selected references A character study Primes in arithmetic progressions Characters of finite abelian groups Proof of the prime number theorem for arithmetic progressions Representations of arbitrary finite groups Characters of arbitrary finite groups Induced representations and examples Applications Generalizations Further remarks 10 Selected references XI Uniform distribution and ergodic theory Uniform distribution Discrepancy Birkhoff's ergodic theorem Applications Recurrence Further remarks Selected references Contents XI1 Elliptic functions Elliptic integrals The arithmetic-geometric mean Elliptic functions Theta functions Jacobian elliptic functions The modular function Further remarks Selected references XI11 Connections with number theory Sums of squares Partitions Cubic curves Mordell's theorem Further results and conjectures Some applications Further remarks Selected references Notations Axioms Index Preface to the revised edition Undergraduate courses in mathematics are colnmonly of two types On the one hand there are courses in subjects, such as linear algebra or real analysis, with which it is considered that every student of mathematics should be acquainted On the other hand there are courses given by lecturers in their own areas of specialization, which are intended to sellre as a prepasation for research There ase, I believe, several reasons why students need more than this Fhst, although the vast extent of mathematics today makes it impossible for any indvidual to have a deep knowledge of more than a small part, it is important to have some understanding and appreciation of the work of others Indeed the sometimes su~prisingintei-relationships and analogies between different branches of mathematics are both the basis for many of its applications and the stimulus for further development Secondly, different branches of mathematics appeal in different ways and require different talents It is unlikely that all students at one university will have the same interests and aptitudes as their lecturers Rather, they will only discover what their own interests and aptitudes are by being exposed to a broader range Thirdly, many students of lnathematics will become, not professional mathematicians, but scientists, engineers or schoolteachers It is useful for them to have a clear understanding of the nature and extent of mathematics, and it is in the interests of mathematicians that there should be a body of people in the coinmunity who have this understanding The present book attempts to provide such an understanding of the nature and extent of mathematics, The connecting theme is the theory of numbers, at first sight one of the most abstruse and irrelevant branches of mathematics Yet by exploiing its many connections with other branches, we may obtain a broad picture The topics chosen are not trivial and demand some effort on the past of the reader As Euclid already said, there is no royal road In general I have concentrated attention on those hard-won results which illuminate a wide area If I am accused of picking the eyes out of some subjects, I have no defence except to say "But what bea~ltif~d eyes!" The book is divided into two parts Past A, which deals with elementary number theory, should be accessible to a first-year undergraduate To provide a foundation for subsequent work, Chapter I contains the definitions and basic propesties of various mathematical structures Fermat prime 184-185 free submodule 198 Fermat's Fresnel integral 161 last th l65,175,20 1,657,659,661-663,667 Frobenius little theorem 129-131, 135, 182 complement 497 Fibonacci numbers 111 conjecture 145 field 27,47,72-74, 127, 146 group 496-497,514 of fractions 102, 116-117 kernel 497 field theory, texts 91 reciprocity theorem 487 finite theorem on division lings 80-82 dimensional 77,79 Fuchsian group 253 field 127,146,271,337,349,450-1,652-4 function 5; 449 field extension 52, 77, 330-331, 336,449 function fields 449-451,459 group 65 and coding theory 45 1,459 set 12 functional equation of finitely generated 73,86,111,193,198,203,639 L-functions 475, 654 Fischer's inequality 279, 283 zeta functions 443, 448, 450, 14 fundamental fixed point 38,41,43 theorems 38, 88 domain 236-238, 392, 406, 575 flex 63 1,632-633,637-638 sequence 30-31, 36-39, 314, 319 flow 537, 554, 556 solution 230-233 formal fundamental theorem of derivative 120, 130, 324 algebra 49-52, 80, 89, 115,480 Laurent series 245,254,307,316,337,419 arithmetic 103-105, 144, 168 power series 113, 145, 321, 335 Furstenberg's theorems 557-560, 564 Fourier Furstenberg-Katznelson theorem 557 integrals 504-505, 14 Furstenberg-Weiss theoreln 558 inversion formula 504 series 92, 160, 441, 505, 514, 522 Galois theory 1, 184 transform 438,441, 504 gamma function 442-443, 458,654,656 fraction 18 Gauss fractional part 520, 551 class number problem 254, 66 1, 667 free invariant measure 552 action 253 map 551-553,556, 563 product 236 sum 158, 159-162,200 subgroup 199,651 Gaussian integer 139-140, 164, 168, 622 Index A15 Gaussian unitary ensemble 446-447 Gauss-Kuz'min theorem 552, 563 GCD domain 102, 104-105, 114, 116-118 group 63-69, 127 generated by reflections 420, 13,514 law on cubic curve 636-637,646,666 gear ratios 18 Gelfand-Raikov theorem 503 group theory, texts 90 general linear group 79, 293, 509 generalized character 496 Haar integral 501-502, 14 geometric principle, stsong and weak 371-372, 381 Hasse-Minkowski theorem 366-370, 380 measure 420,502, 546 tiigonometsic polynomial 86,530 Hadamasd upper half-plane 254 design 291 generated by 68,73,76,106,133,186,389,538 dete~minantproblem 261,284-288,291,300 generating function 624,664 inequality 268-269 generator matrix 261,269-272,291-299, 376,411 matrix 297 Hales-Jewett theorem 561-562, 564 of cyclic group 133 Hall's theorem on solvable groups 496, 514 genus of Hamiltonian system 553, 556 algebraic curve 251,459, 639 Hamming distance 34,297 field of algebraic functions 450,45 Hardy-Ramanujan expansion 628,666 geodesic 242-244,254,452,459,554 Hasse flow 452,459,554, 556 invariant 363, 365, 380 representation of complex numbers 47, 55 series 40 geometiy of numbers, texts 419 Golay code 297-298 golden ratio 209 Good-Churchhouse conjectures 455,459 good lattice point 536 heat conduction equation 596 graph 35 Grassmanil algebra 299 height of a point 639-642 Hensel's lemma 324-5, 327-9, 338 greatest common divisor 97- 102, 104-108 common left divisor 192-193 common right divisor 141 lower bound 22, 26, 29 Hesmite constant 407-408, 41 1,421 normal f o m 202, 391 highest coefficient 13 Hilbert field 359-365, 380 Hasse-Weil (HW) conject~u-e654 Hausdorff distance 413-414 maximality theorem 559 metric 413-414,417-418 Hilbert space 87, 91-92, 502 Ikehara's theorem 437-440,448,452,454,474 image Hilbert symbol 355-360, 367,380 Hilbert's problems, 5th 509-510,514 9th 200-201 10th 253 17th 379, 381 18th 406,420,421 H-matrix 269-270, 284,286 holomorphic function 56, 145, 436, 440, 588 homogeneous linear equations 79, 191 homomorphism of groups 60, 61, 67-68, 648 Lie algebras 11 Lie groups 11 rings 73-74, 116, 129 vector spaces 78 Horner's rule 115 Huxwitz integer 140-141, 147, 62 hyperbolic area 242-243 geometry 241-243,254 length 242 inflection point 631 inhomogeneous Lorentz group 13 injection injectivemap 5, 11-12,78 plane 349,350, 354 hypercomplex number 89 hypergeomeuic function 15 inner product space 82-87, 91,479, 502 integer 12-18, 21 of quadratic field 163-164 hyperreal number 88 integrable 538 in sense of Lebesgue 38, 87,505 in sense of Riemann 385,4 19, 52 1-522 ideal 73-74, 106, 168 class group 175 in quadratic field 169-175,241 of Lie algebra 11 identity element 9, 14,64-65,70,75, 97 map imaginruy part 47 quadratic field 163 incidence matrix 288-291 included indecomposable lattice 409-410 indefinite quadratic folm 238 indeterminate 112 index of quadratic space 347, 348 subgroup 67, 69,486,491-492, 649 indicator function 385, 419, 521, 541 individual ergodic theorem 539 induced representation 486490,493 induction 10 infimum 22,26 infinite order 69 integral divisor 449 domain 72, 102, 12-113 equations 86,92, 261 lattice 410,616 representation for r-function 443, 656 interior 33, 392, 395,402 intersection of modules 191- 192 of sets 3-4, 71 of subspaces 76 interval 32, 34, 75 invarimt factor 197 mean 506 region 553 subgroup 67 subset 558 subspace 477 inverse 14, 19, 64, 72, 177 class 484 element 64-65,72,637,666 function theorem 40-42, 88 map inversion of elliptic integral 590, 593 of order 65-66, 152-153 invertible element of ring 72 matsix of integers 186 measure-pres transf 538,554-555 involutory automorphism 48, 163 irrational number 26,209,520,523 ksationality of 42 1, 116-117 irreducible character 48 1,506 curve 631-633 element 104-105, 168 ideal 171 polynomial 115, 119, 129-130 representation 478-485,503,506 irredundant representation 403 isometric metric spaces 36-37 quadratic spaces 35 isometry 36, 87, 242, 351-353, 406, 498 isomorphism 6, 17, 21, 28 of groups 68 of measure-preserving transformations 555 of rings 74 of vector spaces 79 isotropic subspace 345 vector 345 Jacobi symbol 152-157, 162, 200 Jacobian elliptic functions 602-607 Jacobi's imaginary transfomation 441-442,596 triple product form 594,614,625,627,665 join Jordan-Hiilder theorem 144 Kepler conjecture 42 kernel of group homo~norphism 1, 68 linear map 78 representation 494 ling homomorphism 73-74 Kewaire-Milnor theorem 90 Kingman's ergodic theorem 563 kissing number 412, 421 K-point, affine 629 projective 630,635 Kronecker approximation theorcm 524,562 Kronecker delta 482 field extension theorem 52 product 270,272,29 1,298,477 Lebesgue measure 38,385,537,543-545,552 Leech lattice 41 1-412,421 Lefschetz fixed point theorem 88 left BCzout identity 141 Lagrange's theorem coprime matrices 193 on four squares 140-142,253,621-622 coset 67 on order of subgroup 67, 129, 133 divisor 192 Landau order symbols 225,429,5 Legendse Landau's theorem 472 interchange property 593 Landen's transformation 606, 613-4, 647-8 nomal form 575 Langlands program 20 1, 666 polynomials 86 Laplace transform 437,458,472 relation 584 lattice 99, 144; 164, 391-392,409 sy~nbol 156, 158, 173, 271, 357 in locally compact group 564 theorem on ternasy quadratic forms 366-367 packing 408, 421 lemniscate 571, 585, 614, 615 packing of balls 408,411, 421 less than point 386, 391 L-function 470, 513, 653-657, 661 translates 395-396 Lie Laurent algebra 10-514 polynomial 115, 25 group 509-514 series 56,245, 307, 419,436 s~lbalgebra 10 law of subgroup 511 iterated logarithm 455,459, 563 limit 28, 35, 314 Pythagoras 21, 84-85, 126 linear algebra, texts 91 151,156-9,162,175,200-1,367 quad rec linear trichotomy 9- 10, 15, 25 code 297,451 least combination 76 common multiple 98-102, 104 differential system 199 common right multiple 192 Diophantine equation 106, 185-6, 190-1 element 10 fractional transfn 209,242,576,609,613 non-negative residue 124, 134 map 78 upper bound 22-23,26,29 systems theory 203 least upper bound property (P4) 23,26,3 transfomation 78 Lebesgue measurable 38,87 lineasly dependent 76-77 linearly independent 76-77 Linnik's theorem 476 Liouville's integration theory 614 theorem in complex analysis 89,594 theorem in mechanics 553 Lipschitz condition 533 Littlewood's theorem 446, 458 LLL-algorithm 19 L h o r m 84 local-global principle 37 1-372,38 locally compact 33 group 420, 501-505, 514-515, 564 topological space 501 valued field 334-337 locally Euclidean topological space 10 logarithm 45,428 lower bound 17,22,26 limit 29 triangular matrix 268 Lucas-Lehmer test 182-183,202 Mahler's compactness theorem 418,42 map mapping 4-5 Markov spectrum 244-245,254 triple 244-245 ma-siage theorem 91 Maschke's theorem 478,499 Mathieu groups 293,297,298 'matrix' 187, 193 matrix theory, texts 91, 300 maximal ideal 74, 171, 15, 320 maximal totally isotropic subspace 346 Mazur's theorem 65 mean motion 530,562 measurable function 35,538 measure-pres transf 538-543, 550-557 measure theoly, texts 563 measure zero 35,554 meet Mellin transform 655 MCray-Cantor construction of reals 22, Merkur'ev's theorem 380 mesomorphic fn 56, 307, 588, 615-616 Mersenne prime 182-183,202 Mertens' theorem 428 method of successive approx 38, 43,45 metric space 33-39, 84,297, 13 Meyer's theorem 366, 370 minimal basis 200 model 652,663,666 vector 404,406 mini~numof a lattice 404,406,407,418 Minkowski's theorem on discriminants 388,419 lattice points 386-388,397, 420 linear forms 386 successive minima 398-400,419 minor 197 mixing transformation 552 Mobius function 180,453-455,459 inversion formula 180,202 modul as elliptic curve 655-657, 661, 667 form 301, 624, 655-6, 661, 666 modular function 608-613,616 group 235-238,609 transformation 234 module 186, 193, 194, 198,202-203 modulo m 124 monk polynomial 113, 175,201, 307 monotonic sequence 29-31 Monster sporadic group 30 Montgomely's conjecture 446-447,458 Mordell conjecture 249,251, 657,667 Mordell's theorem 203, 639,646-651 multiple 97 multiplication 70 by a scalar 74 of integers 14-15 of natural numbers 8-9 of rational numbers 19 multiplicative function 178-179,202 group 72, 133, 146, 342 inverse 19 Nagell-Lutz theorem 65 1,660 natural logarithm 45,428 number 5-12, 17 nearest neighbour conjecture 447 negative definite quadratic space 347 index 348 integer 16 neighbowhood 39 Nevanlinna theory 249,254 Newton's method 324, 338 node 634,635,653,654 non-archimedean abs value 306,308,318-322 non-associative 62-63, 90 nondecreasing sequence 29-30 nondegenerate lattice 391 non-Euclidean geometry 241-243,254 line 241-242 triangle 243, 10 nonincreasing sequence 29-30 non-negative linear functional 50 nonsingular cubic curve 635 linear transformation 78 matrix 265 point 629, 630 projective cuive 450 projective variety 45 quadratic subspace 343 norm of complex number 139 continuous function 34 element of quadratic field 123, 163 ideal 447 integral divisor 450 linearnap 40 n-tuple 33-34 octonion 62-63 prime divisor 450 quaternion 57-61, 141 vector 83,3 17,400 normal form for cubic curve 632-633,637-638 frequencies 498 modes of oscillation 498 no~mal number 546,548,563 subgroup 67-68, 91,488,495 vector 546-548,563 no~m-Euclideandomain 123 normed vector space 17,335,400 n-th root of complex number 51,55,89 positive real number 27 a-tuple 4, 33, 75 nullity of linear map 79 nullspace of linear map 78 numbers 1, 87 number theory, texts 144 numerical integration 533, 536, 562 octave 61 octonion 61-63, 90, 512 odd permutation 65-66, 152, 262, 266 one (1) 6, 70 one-to-one correspondence open ball 33, 39, 401 set 33,50-51 operations reseasch 91 Oppenheim's conjecture 379,38 order in natural numbers order of element 69, 132 group 65, 127 Hadamard matrix 269 pole 56 projective plane 289 ordered field 27,30,47,88,91,326,347-8,361 ordinay differential equations 43-4,88,586-8 osientation 263,407 Ornstein's theorem 555 orthogonal basis 86, 344, 394 orthogonal complement 343 group 509 matsix 60,277 set 85 sum 343,409 vectors 84-85, 343 orthogonality relations 469,482-483,485,506 orthonormal set 85-86 Oseledets ergodic theorem 563 Ostrowski's theorems 311, 332, 338, 366 packing 395, 396,421 p-adic absolute value 306 integer 321, 323, 335 number 22,316,321,323,336,356,366,505 pair correlation conjecture 446, 458 Paley's construction 27 1-272, 297, 41 pasallelogram law 84, 91, 642, 645 pasallelotope 268, 392 pasametrization 59,251-252, 255, 635, 639 Parseval's equality 86-87, 394-395 partial fractions 573 order 99 quotient 212,221, 247, 552 partition of positive integer 624-628 set 4, 67 pastition theo~y,texts 664 A22 Index Pascal triangle 110 path-connected set 51,507,511 Peano axioms 6,87 Pel1 equation 167,228-234,252-253 for polynomials 248,254 pendulum, period of 570 polynomial part 246 ring 102,121 polytope 403,420 Pontryagi~l-vanKampen theorem 504 positive Pkpin's test 185 percolation processes 563 perfect number 181-183, 202 period of continued fraction 224-5,229-230 periodicity of continued fraction 224-225,243,252 elliptic functions 590-593,604, 616 index 348 integer 15-17 measure 502 rational number 20-2 real number 22,26 semi-definite matrix 274, 279 positive definite matrix 274, 279 quadratic form 238 quadratic space 347 rational function 379 power series 45, 52-55 primality testing 144-145 exponenth1 function 53-54 permutation 65, 152,266 perpendicular 84 Perron-Frobenius theorem 553 Pfister's multiplicative forms 379 pi (n) 54-55, 217, 252,428, 585 Picard's theorem 616 pigeonhole principle 12, 65 Plancherel theorem 504 Poincark model 241-242,254 recurrence theorem 556 point 288, 292, 629, 630 at infinity 630, 633 pointwise ergodc theorem 539 Poisson summation l61,441,458,504,616 polar coordinates 55, 571 lattice 392 pole of order 11 56 poles of elliptic functions 603 polynomial 112- 121 psime divisor 449 element 104-105, 168 ideal 171-174,447-448 ideal theorem 448,457, 458 number 103-104 prime no th 429-43 1,433-440,454,457-9 for arithmetic progressions 457,466,469-475 primitive Dirichlet character 475 polynomial 117 quadratic form 240 root 133-135, 145,448-449 root of unity 129-130, 133 principal axes transforn~ation 278,299-300 principal character 467 Index principal ideal 106, 169 properly isomo~phic 407 domain 108,111-112,114,121,123,194-198public-key cryptography 145 principle of the argument 10-611 probability measure 538 space 538 theory 35, 88,455,459, 666 probletn, 3x + 563 problem of moments 255 product formula for theta functions 596-597 formula for valuations 12 measure 549 of ideals 169 of integers 14 of lineax maps 79 of natural numbers of rational numbers 19 of representations 477 of sets projective completion 630, 632 conic 630 cubic 630 equivalence 63 1,632 line 630 plane 289, 291, 376, 381 plane curve 629 space 61 proper divisor 104 subset properly equivalent complex numbers 214-215,234 quadratic forms 239 Puiseux expansion pure imaginary complex number 47 quaternion 58-59 Pythagoras' theorern (or law) 21, 84-85, 126 Pythagorean triple 126,252 q-binomial coefficient 110,664-665 q-difference equation 626 q-hypergeometric series 665 q-integral 664 quadratic field 123, 145, 163-175,201,241,253 form 238-241, 342, 644 irrational 222-226, 240, 243-244, 248, 615 nature 151, 156 non-residue 131-2, 142, 151, 155-6, 448 polynomial 49,329 residue 131-132, 142, 151, 155-156, 326 space 342-355,380 quadratic spaces, texts 379-380 quantum gsoup 665 quartic polynomial 88 quasicrystal 420 quasiperiodic tiling 420 quaternion 56-62, 80-81, 89, 140-142, 621 quaternionic analysis 89 QuillenSuslin theorem 202-203 quotient 18, 21, 105 group 67 11ng 73-74, 125, 447 space 243, 244,253 Riidstrom's cancellation law 414,421 Ramanujan's tau-function 452,459 random matrices 446-447,458,563 range of linear map 78 rank of elliptic curve 651-652,655 linear map 79 rational function 102, 247, 306, 449 number 18-21,212, 323 transformation 638 red analysis 32, 88 number 26-32 part 47 quadratic field 163 reciprocal lattice 392 reciprocity for Gauss sums 160 reculrence for number of partitions 625 recursion theorem 6-7 reduced automorphism group 294 lattice basis 419 quadratic form 240 quadratic irrational 223-226 reducible curve 631 polynomial 329 representation 478 reducibility criterion 329 Reed-Muller code 298 refinement theorems 101, 144 reflection 61, 351-352 reflexive relation regular prime 175-176 regular reprn 476-477, 483, 503, 506 relatively dense set 402 prime 99, 193 relevant vector 404 remainder 105 theorem 16 replacement law 124 representation of compact group 506 finite group 476-480, 506, 513 group 476,513, 514 locally compact group 503 representative of coset 66,90 residue field 322 representatives, distinct 91 represented by quadratic form 344-346 residue 56,435,453 class 124, 466 field 320-322, 449 resolution of singularities 638-639 restriction of map Ribet's theorem 663 Riemann integrable 385,419, 521-522, 524 normal form 574-578, 586,636 surface 253 zeta fn 430,434-437,443-447,453-454,458 Riemann hypothesis 444-446,454-455,459 for algebraic varieties 450-451, 459 for elliptic curves 653, 657,666-667 for function fields 450-451,459, 667 Riemannian manifold 452, 554, 556 Riemann-Lebesgue lemma 439 Riemann-Roch theorem 400,420,450 Riesz representation theorem 502 Riesz-Fischer theorem 87 right coset 66-69 multiple 192 vector space 79 ring 70-74, 79, 113, 124 set of representatives 322, 337 shift map 550,55 1, 561 Siegel's formula 394-395 lemma 399,419 modular group 254 theorem on Diophantine eyns 25 1,255,667 sigma algebra 502,537-538 ring theory, texts Rogers-Ramanujan identities 626-628,665 root 116-117,255, 324 lattice 410-411, 421 sign of a permutation 66, 152, 266 signed permutation matrix 282, 286-287,293 simple associative algebra 80 Roth's theorem on alg nos 143,248-251,254 basis 410-41 ruler and compass constructions 184, 202 group 67, 293, 301, 495 Lie algebra 11-514 scalar 75 Lie group 90,293 schemes 666 pole 56, 435,436,448, 452, 453 Schmidt's orthog process 85-86, 268 ring 73 Schmidt's simply-connected 1,506 discrepancy theorem 535, 563 covering space 1, 12 subspace theorem 249,254 Lie group 510, 12 Schreier's refinement theorem 144 Schur's lemma 479-480 Schwarz's inequality 34, 83, 273, 436, 525 self-dual lattice 392, 394, 399 semidisectproduct 496 semigroup 88 semi-simple Liealgebra 511, 514 Lie group 12, 14 semi-stable elliptic cuive 654, 657,663 Serre's conjectuse 202-203 &-conjecture 663 set 2-5, 71 simultaneous diagonalization 279,280,498 singular matrix 265,267 small divisor problems 255 oscillations 300, 497-498 Smith nomal foini 195-199,203 sojourn time 541 solvable by radicals 46, group 495,496 Lie algebra 14 spanned by 76 special 61 linear group 235,267 A26 Index special orthogonal group 61,508-509,512 subspace 75-78 successive unitary group 61,506-509,5 12 spectrometry 276, 299 spherical trigonometry 616 sporadic simple group 293,301,412 square 16 class 342, 343, 646-647 design 290-291,300,376 approximations 38,43 minima 398,419 successor sum of linear maps 79 modules 191 nat~ralnumbers square-free element 105 integer 105, 163 polynomial 120-121, 130 square-norm 401,404 square root of complex number 45,49 positive real number 27,29 square 2-design 290-291, 300, 376 star discrepancy 53 1,536 Steiner system 292 step-function 521 Stieltjes integral 255,432,436-437,458 Stirling's formula 386, 443 Stone's representation theorem 71, 87 Stone-Weierstrass theorem 562 strictly proper part 246 strong Hasseprinciple 371-372 triangle inequality 35,38, 306 structure theorem for abelian groups 198-199,203 for modules 198-199 subadditive ergodic theorem 563 subgroup 65 subset 2,71 points of elliptic curve 636-637, 666 representations 478 subspaces 76 sums of squares: 63, 90, 147, 624, 664 two 125,139-l40,232-3,253,287-8,622-4 three 125, 140, 372-373 four 59, 140-142,253, 621-622 for polynomials 377-379, 38 for rational functions 379, 38 supplements to law of quad rec 156, 367 supremum 22,26 sulface area of ellipsoid 57 1-572 of negative cumature 452,459, 554, 556 surjection surjectivemap 5, 11-12,78 Sylvester's law of inertia 348 symmetric difference 1,538 group 65-66,266,490-493 matrix 278, 342-345, 352 relation Riemannian space 254 set 386,400 syrnrnehic 2-design 290 symmetry group 406,498 symmetry operation 498 symplectic matrix 254, 509 systems of distinct representatives 91 transcendental number 201,66 1,667 transformation formulas for elliptic functions 588-589,605-606,616 Szemeredi's theorem 557-558 for theta functions 441-442, 596, 598 transitive law relation translation 406 of torus 544, 555 transpose of a matrix 265,267 tsiangle inequality 33,306 triangular matrix 268 trichotomy law 9-10 higonometric functions 53-5589 polynomial 86, 522-523, 562 bivial tangent space 510,554 to affine curve 629 to projective curve 630 taxicab number 136 Taylor series 56,437,472 t-design 292-293,296-298 theta fn 442,595-601,607-8,612-3,621-4 of lattice 616-617 tiling 238,392,395,402,404,406,420,575 topological entropy 452 field 314 group 501, 509-510, 514 topology 33, 13 torsion group of elliptic cusve 651, 660 subgroup 69, 199 submodule 198 torsion-free 406,420 torus, n-dimensional 509, 543-545 total order 9,22,28 variation 533-534,562 totally isotropic subspace 345347,350 totient function 128 trace of matrix 480 quaternion 57-58 transcendental element 449 absolute value 306, 19 character 467 representation 476 ring 71 TW conjecture 656-657, 661, 663, 667 twin prime 455-457,459 twisted L-function 656 2-design 288-292, 300 type (A) Hilbert field 361-363 type (B) Hilbert field 361-363, 365 ultrametsic inequality 306, 318, 324 uniformly distributed mod 520-530 uniform distribution, texts 562 unifo~mizationtheorem 253 union of sets 3-4,7 unique factorization domain 105 unit 72, 102, 127, 167-168, 177 unit circle 54-55 unit tangent bundle 554 unitary group 509, 12 matrix 61, 506 representation 479, 503, 506 symplectic group 509,s 12 universal quadratic form 345,348 upper bound 22-23,26 density 557 half-plane 234,241-243,595,608,612 limit 29 triangular matrix 268 valuation ideal 320-322 ring 102, 320-322 valuation theory, texts 338 value group 306-307,319, 322 valued field 305-309 van der Corput's difference theorem 526,529 sequence 535-536 van der Waerden's theorem 558, 561,564 vector 75 space 74-82 vertex of polytope 403 volume 385 van Mangoldt function 434-435 Voronoi cell 401-406,410,421 of lattice 403-406,414-418,421 Voronoi diagram 420 Waring's problem 142-144, 147 weak Hasse principle 37 1-372 Wedderburn's theorem on finite division rings 146 simple algebras 80 Weierstrass approxn theorem 86, 522, 562 weighing 273-276, 299 matrix 273,275,299 weight of a vector 297-298 Weil conjectures 450-452,459 Weyl's criterion 522-523 Wiener's Tauberian theorem 43 1, 457-458 Wiles' theorem 657, 663, 667 Williamson type 272 Wilson's theorem 130-131 Win cancellation theorem 352-353,355, 371 chain equivalence theorem 363-364 equivalence 355 extension theorem 352-354 ring 355, 379, 380 zero 14,70 zeros of elliptic functions 603 zeta function 430,434-437,443-447,458 generalizations 452,459 of function field 450 of number field 447-448,458,475 ... States of America For Jonathan, Nicholas, Philip and Stephen Contents Part A Preface I The expanding universe of numbers Sets, relations and mappings Natural numbers Integers and rational numbers... whenever aRb; transitive if aRc whenever aRb and bRc It is said to be an equivalence relation if it is reflexive, symmetric and transitive If R is an equivalence relation on a set A and a E A, the... I have also corrected a few misprints, made many small expository changes and expanded the index Although I have made a few changes to the references, I have not attempted a systematic update