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Cohen H.A. — A Course in Computational Algebraic Number Theory
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Название: A Course in Computational Algebraic Number TheoryАвтор: Cohen H.A. Аннотация: A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1996Количество страниц: 563Добавлена в каталог: 28.05.2005Операции: Положить на полку |
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MLLL algorithm 96 MOD 7 Modular equation 386 Modular forms 234 390 Modular functions 379 Modular invariant 377 Modular multiplication 4 Module 188 Module,denominator 188 Modules, product of 189 Montgomery, P. 5 429 489 492 Morain, F. 445 471 474 Mordell, L. 375 MPQS 490 MPQS, self-initializing 494 Multi-precision 2 Nagao, K. 394 Narrow class group 228 Neumann, W. 102 Newton polygon 313 Newton’s Formulas 163 Newton’s method 38 45 NFS 495 Non-split multiplicative degeneracy 373 Norm of a fractional ideal 187 Norm of an element 162 Norm of an ideal 182 Normal closure 157 NP-complete 103 NUCOMP 247 NUDUPL 247 Number field 154 Number field, primitive 335 Odlyzko, A. 254 465 Oesterle, J. 295 Olivier, M. 171 175 254 288 313 329 333 352 513 Order 181 Order of a group element 24 Orthogonal basis 82 Orthogonal matrix 81 p(z) 368 p-adic factorisation 363 p-adic regulator 300 p-adic valuation 186 p-maximal 303 p-radical 303 PARI 2 508 Partial quotients 22 Period lattice 368 398 Permutation matrix 50 PID 183 Pivot 48 65 Place, finite 187 Place, infinite 187 Place, of a number field 187 Pohst, M. 96 304 Pollard, J. 426 439 495 Polya — Vinogradov inequality 301 476 Polynomial time 2 Pomerance, C. 445 465 481 490 501 Powering algorithms 8 42 466 Powers, R. 478 Powersmooth number 439 Primality certificate 470 Prime element 114 Prime form 252 Prime ideal 184 Prime ideal theorem 215 Prime number theorem 215 Primitive algebraic integer 274 Primitive algebraic number 497 Primitive element 155 Primitive element problem 181 Primitive ideal 225 Primitive part 116 Primitive polynomial 116 Primitive quadratic form 225 Primitive root 24 Principal ideal 183 287 Principal ideal domain 114 183 Principal minors 53 Probabilistic algorithm 2 Product of ideals 182 Projective geometry over 485 Pseudo-division 112 Pseudo-prime 422 q 378 QS 490 Quadratic form 79 225 Quadratic form, positive definite 80 Quadratic reciprocity law 27 Quer, J. 297 Rabin, M. 421 Ramification index 197 Ramified prime 197 Rank 66 Rank of an elliptic curve 375 Reduce 2 507 Reduced basis 84 Reduced ideal 300 Reduced quadratic form 231 262 Reduction of quadratic forms 243 Regular primes 209 Regular representation 160 Regulator 211 Regulator, elliptic 411 Relative extensions 329 Residual degree 197 Resolvent polynomial 323 Resultant 119 Ribet, K. 392 Roots of unity 209 Row vector 7 Rubin, K. 394 Rumely, R. 445 Schnorr, C. 91 481 Schonhage, A. 3 150 Schoof, R. 32 405 469 Separable extension 166 Serre, J.-P. 392 Shanks, D. 32 241 247 251 279 288 433 434 Shimura, G. 392 Side exit 65 Signature 155 Silverman, J. 367 Simath 508 Singular number 501 Size of a polynomial 168 Small prime variation 494 Smith normal form 67 75 Smooth number 439 SNF 67 75 Solovay, R. 421 Spar 481 Sparse matrix 254 480 Sparse representation 109 Special subset 486 Split multiplicative degeneracy 373 Splitting 419 Square form 434 Square root in 38 Square root modulo p 31 Standard fundamental domain 231 Standard representation 159 Stark, H. 382 Stickelberger, L. 167 198 Strassen, V. 3 421 Strong pseudo-prime 422 Sturm, J. 155 Sub-exponential algorithm 2 Sub-resultant algorithm 118 122 Subfield problem 174 Supersingular 382 Supplement 61 Swinnerton-Dyer, H. 392 Sylvester’s matrix 120 Symmetric function 162 Taniyama — Weil conjecture 391 Taniyama, T. 391 Tate — Shafarevitch group 393 Tate, J. 407 Taylor, R. 392 Titanic numbers 471 Torsion subgroup 66 375 Totally complex 155 Totally real 155 Trace 162 Transitive 323 Trial division 419 Triple Jacobi sum 460 Tschirnhausen transformation 324 Two element representation 193 Ubasic 2 508 UFD 114 Unique factorization domain 114 UNIT 114 209 Unramified prime 197 Upper half-plane 378 Vallee, B. 84 Valuation 201 van der Hulst, P. 466 Weber class polynomial 417 Weber functions 474 Weierstrass equation 370 Weierstrass equation, minimal 370 406 Weil Conjectures 387 Weil curve 392 Weil, A. 375 387 391 Wiedemann, D. 254 Wieferich congruence 459 Wiles, A. 392 Williams, H. 279 285 Winter, D. 445 Wolstenholme’s Theorem 476 Zagier, D. 216 234 236 385 394 Zassenhaus, H. 127 304 Zeta function of a number field 214 Zeta function of a variety 388
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