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Cohen H.A. — A Course in Computational Algebraic Number Theory
Cohen H.A. — A Course in Computational Algebraic Number Theory



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Название: A Course in Computational Algebraic Number Theory

Автор: 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: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1996

Количество страниц: 563

Добавлена в каталог: 28.05.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$2^k$-representation      10
$j(\tau)$      378
$v_p(I)$      186
$\bar{\mathbb{Q}}$      153
$\Gamma_0(N)$      390
$\mathbb{Z}$-module      66
$\mathbb{Z}_K$      154
$\mathcal{H}$      378
$\mathcal{I}(K)$      208
Abelian group      66
Addition chain      11
Addition theorem      370
Additive degeneracy      373
Adeles      188
Adjoint matrix      54
Adleman, L.      445 471 501
Affine subspace      486
Algebraic integer      153
Algebraic number      153
Algorithm      1
Ambiguous cycle      270 434
Ambiguous form      255 433 434
Antwerp IV      367
Approximation theorem      192
Archimedean valuations      187
Artinian rings      303
Atkin, O.      32 247 252 445 471 481
Axiom      2 507
Baby step giant step      240
Bach, E.      34 254
Bad reduction      373
Bareiss, E.      52
Berge, A.-M.      175 329
Berlekamp, E.      130
Bernardi, D.      305 382 394
Bignum      2 509
Binary quadratic form      225
Birch and Swinnerton — Dyer conjecture      393
Birch, B.      392
Birthday paradox      480
Bit operations      1
Bosma, W.      466
Brauer — Siegel theorem      216
Brent, R.      420 427 429 441
Buchmann, J.      288 303 315 352
Buhler, J.      501
Canfield, E.      481
Canonical height      411
Canonical height pairing      411
Cantor, D.      127
Carayol, H      390
Carmichael numbers      421
Cartan decomposition      107
Cartan, E.      107
CEILING      7
cfrac      477
CHARACTER      448
Characteristic polynomial      53 162
Chinese remainder theorem      19
Cholesky decomposition      104
Chudnovsky, D.and G.      490
Cl(K)      208
Class group      207 228
Class number      208
Codifferent      205
Coefficient explosion      5 112 114
Cohen, H.      168 254 288 296 352 445
Collins, G.      118
Column echelon form      59
Column vector      7
Comatrix      54
Compact representation of units      279 285
Complementary error function      238
Completely split prime      197
Complex multiplication      381
Compositeness test      419
composition      243
Conductor      224
Conductor-discriminant formula      168
Congruent number      376
Conjugate vector representation      161
Conjugates      154
Content      116
Continued fraction      21 265 271 426 478
Coordinate recurrence method      254
Coppersmith, D.      480 504
Coprime elements      116
Coprime ideals      182
Couveignes, J.-M.      405
Cremona, J.      394 417
Cycle of reduced forms      262
Cyclotomic field      446
Davenport, H.      462
de Bruijn, N.      481
de Weger, B.      93
Dedekind domain      185
Dedekind zeta function      214
Dedekind, R.      305
Dedekind’s eta-function      416
Deep insertion      91
Degenerate elliptic curve      373
Degree of a prime ideal      197
Deligne, P.      387
Denominator of a module      74
DERIVE      2 507
Determinant      52
Determinant of a lattice      80
Diaz y Diaz, F.      168 254 288 313 352 363
Different      205
Dirichlet character      448
Dirichlet, P.      211
Discriminant of a number field      166
Discriminant of a polynomial      119
Discriminant of a quadratic number      384
Discriminant of an n-element set      165
Distance function      279
Distinct degree factorization      126
Divisibility (in a ring)      114
Division polynomials      405
Double large prime variation      494
Doubly periodic function      368
Dual isogeny      380
Duke, W      298
Dullmann, S.      252 254
Dwork, B.      388
Early abort strategy      259 480
ECM      487
Eichler, M.      236
Eisenstein polynomial      315
Elementary divisor      76
Elementary operations      48
Elimination      48
Elkies, N.      405 471
Elliptic curve over K      369
Elliptic function      368
Elliptic integral      367 397
Elliptic logarithm      398
Enlarging procedure      304
Equivalence of quadratic forms      225
Erdos, P.      481
Euchner, M.      91
Euclidean Domain      114
Euclid’s algorithm      12
Euler product      250
Expected running time      2
Exponential time      2
Factor base      260 478
Fast multiplication methods      3
Fermat number      424 495
Fermat’s Last Theorem      151 208 392 459
Fermat’s Little Theorem      421 439 450
Fermigier, S.      394
Field membership problem      179
Fincke, U.      105
FLOOR      7
Floyd, R.      427
FLT      151 208
Fractional ideal      183
Frobenius homomorphism      309
Functional equation for elliptic curves      390
Functional equation for number fields      215
Functional equation for quadratic fields      238 266 267
Functional equation, sign of      391
Fundamental discriminant      224
Fundamental domain      368
Fundamental units      210
Galois closure      157
Galois group      157 322
Gap      508
Gauss sum      448
Gauss, K.F.      52
Gaussian elimination      48
Gaussian pivoting      48
Gaussian reduction      23
Gauss’s Lemma      116
GCD      7 115
Generalized Riemann hypothesis      34
Genus field      474
Germain, S.      151
GMP      2 509
Goldfeld, D.      216 234
Goldwasser, S.      445
Gram matrix      80
Gram — Schmidt orthogonalization      82
Greatest common divisor      7 12 115
GRH      34
Gross, B.      216 234 385 394
Group ring      446
h(K)      208
H(n)      234
Hadamard’s inequality      51 82
Hafner, J.      69 70 77 252
Hashing      299
Hasse — Weil L-function      389
Hasse, H.      373 462
Height      411
Hensel’s Lemma      137
Hermite normal form      67
Hermite normal form of a $\mathbb{Z}$-module      67
Hermite normal form of a matrix      67
Hermite, C.      198
Hermite’s constant      334
Hessenberg form      55
Hilbert class field      384 416
Hilbert class polynomial      415
HNF      67
HNF-basis      189
Huang, M.D.      445 471
Hurwitz class number      234
Ideal      182
Ideal, class      208
Ideal, equivalence      207
Ideal, intersection      207 219
Ideal, inversion      204
Ideal, product      190
Ideal, representation      188 190
Ideal, two-element representation      192
Ideal, valuation      201
Idele class group      209
Ideles      188
Image of a matrix      58
INDEX      167
Inert prime      197
Inessential discriminantal divisor      199 364
Infinite prime      198
Infrastructure method      279
Integral basis      166
Integral domain      114
Integral ideal      183
Integral kernel      74 98
Integrally closed      185
Intelligent Gaussian elimination      480
Intelligent Hermite reduction      254
Inverse image      60
Irreducible element      114
Irreducible polynomial      124
Isogeny      379
Isomorphism problem      179
Iwasawa decomposition      83
j(E)      377
Jacobi sum      448
Jacobi symbol      28
Kant      508
Karatsuba, A.      3
Kernel of a matrix      57
Kilian, J.      445
Knuth, D.      298
Kodaira type      407
Kolyvagin, V.      394
Kouya, T.      394
Kraitchik, M.      478
Kronecker symbol      28
Kronecker, L.      211
L(p)      109
L(X)      254
L-function      266 388 389
L-series      237
LaMacchia, B.      89 254
Large prime variation      258 480
Laska, M.      409
Lattice      23 80
Lattice, determinant of      80
Legendre symbol      27
Legendre symbol, generalized      219
Legendre, A.      478
Lehman, S.      425
Lehmer, D.H.      13 423 443 478
Lenstra, A.K.      84 141 494 495
Lenstra, H.W.      84 141 184 201 296 298 303 315 320 419 442 445 466 481 484 503
Leopoldt’s conjecture      216
lg      7
Lisp      2
LLL algorithm      87
LLL algorithm, integral      94
LLL-reduced basis      85
Logarithmic embedding      210
Louboutin, S.      301
Lovasz, L.      84 141
Lucas — Lehmer test      443
Lucas, E.      443
LUP form of a matrix      50
Macsyma      2 507
Magma      2 508
Manasse, M.      494 495
Manin, Y.      392
Manin’s constant      392
Maple      2 507
Martinet, J.      217 329
Mathematica      2 507
Matrix representation      160
Maximal ideal      184
Maximal order      186 303
Mazur, B.      375
McCurley, K.      69 70 77 252 288
Mersenne number      424 443 495
Mestre, J.-F.      394 418
Mignotte, M.      134
Miller, G.      421
Minimal polynomial      153
Minkowski, H.      198
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