Number Theory: An Introduction to Mathematics

586 XIII Connections with Number Theory [50] J.H. Silverman,The arithmetic of elliptic curves, Springer-Verlag, New York, 1986. ...

Notations........................................................ ∈,/∈,=,=,∅,⊆,⊂,∪,∩, 2 B\A,Ac,3 A×B,An,aRb,3 Ra,f:A→B,f(a),f(A ...

588 Notations B,74 ^2 ,L^2 (I),75 ∗R,76 b|a,b
a,×,(a,b),83 [a,b], 84 a∧b,a∨b,85 (a 1 ,a 2 ,...,an),[a 1 ,a 2 ,...,an], 86 K[t], ...

Notations 589 |f|∞,vp(f),|f|p, 262 K((t)), 263 F, 270, 271 Qp,‖a‖, 271 R,M,U, 274 Zp, 275 π,k, 276 f(x), 280 F×,F×^2 ,(u,v), 292 ...

590 Notations ξ,{ξ},I,φα,β(N), 448 χα,β, 449 e(t), 450 φa,b(N),Id,{x},m·x, 452 D∗N=D∗N(ξ 1 ,...,ξN),φα(N), 459 δN, 462 D∗N(x 1 ...

Axioms.......................................................... (A1)–(A3), 7 (A4),(A5), 12 (AM1), 7 (AM2), 13 (B1), (B2), 465 ( ...

Index............................................................ abelian group, 55, 79, 172, 196, 400, 434, 441, 555, 556 Lie a ...

Index 593 arithmetic-geometric mean, 502–509, 512, 535–536 arithmetical function, 152–154, 175 arithmetical functions, 153, 175 ...

594 Index cancellation law, 7, 76, 83, 147 canonical height, 561–565 Cantor’s construction of reals, 18, 26, 269 Carath ́eodory ...

Index 595 of symmetric matrices, 293 subgroup, 210 congruent, 106 congruent numbers, 575–578, 584 conic, 549, 550 conjugacy clas ...

596 Index polynomial, 96–97 representation, 410, 414 De Morgan’s laws, 3 dense, 17–18, 275, 447 sequence, 448, 452 subset of met ...

Index 597 elliptic curves, 555–558, 569–574, 582–583 functions, 509–516, 525–530 elliptic integral, 496–503, 536–537 of first ki ...

598 Index Fermat number, 160–161, 175 Fermat prime, 160–161 Fermat’s last theorem, 151–152, 175, 579– 580, 584 little theorem, 1 ...

Index 599 generator matrix, 255 of cyclic group, 114 genus of algebraic curve, 216, 395, 558 field of algebraic functions, 387, ...

600 Index hypergeometric function, 537 hyperreal number, 76 ideal, 63, 90–91, 145, 148 class group, 151 in quadratic field, 146– ...

Index 601 of measure-preserving transformations, 482–483 of rings, 64 of vector spaces, 68 isotropic subspace, 295 vector, 295 J ...

602 Index subgroup, 441 limit, 24, 30, 269 linear code, 255, 388 combination, 65 differential system, 172–173 Diophantine equati ...

Index 603 function, 531–536, 538 group, 202–205, 531 transformation, 201 module, 161, 166–169, 171–172, 175 modulom, 106 monic p ...

604 Index correspondence, 4 open ball, 28, 33, 342 set, 28, 43 operations research, 78 Oppenheim’s conjecture, 324, 325 order in ...

Index 605 part, 211 ring, 87, 104 polytope, 343, 358 Pontryagin–van Kampen theorem, 435 positive index, 297 integer, 13–14 measu ...