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Everest G., Ward T. — An Introduction to Number Theory
Everest G., Ward T. — An Introduction to Number Theory

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Название: An Introduction to Number Theory

Авторы: Everest G., Ward T.

Аннотация:

The book aims to take readers to a deeper understanding of the patterns of thought that have shaped the modern understanding of number theory. It begins with the fundamental theorem of arithmetic and shows how it echoes through much of number theory over the last two hundred years. One of the main strengths of this book is the narrative. Everest and Ward present number theory as a living subject, showing how various new developments have drawn upon older traditions.

The authors concentrate on the underlying ideas instead of working out the most general and complete version of a result. They select material from both the algebraic and analytic disciplines and sometimes present several different proofs of a single result to show the differing viewpoints and also to capture the imagination of the reader and help them to discover their own tastes. They also cover important topics of significant interest, eg. elliptic functions and the new primality test, which are often omitted from other books at this level.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2005

Количество страниц: 294

Добавлена в каталог: 19.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abel      5
Abel, limit theorem      210 223
Abel, Summation Formula      219 235
Abelian theorem      224
Abelian variety      97
Additive reduction      241
Adeles      224
Agrawal — Kayal — Saxena algorithm      266 277
al-Haytham      5 62
al-Haytham, Theorem      32 48 60 257
al-Haytham, Theorem, Clement's extension      33
al-Haytham, Theorem, Gauss' generalization      32
Algebraic integers      84 226
Algebraic integers, prime ideals      85
Algebraic integers, quadratic field      84 86 91 228 231 232
Algebraic integers, units      77 85
Analytic continuation      175
Analytic continuation, Dedekind zeta function      229 230
Analytic continuation, L-function      219
Analytic continuation, L-function of elliptic curve      242
Analytic continuation, Riemann zeta function      175 178 185
Analytic function      170
Analytic function, Morera's Theorem      174
Analytic function, uniform convergence      173
Ankeny      268
Arithmetic function      60
Arithmetic function, completely multiplicative      60
Arithmetic function, convolution      164
Arithmetic function, multiplicative      60
Artin      5
Artin, conjecture      65
Associative law      97
Bachet      5
Bachet, equation      93 118
Bernoulli number      203
Bernoulli number, regular prime      83
Bertrand's postulate      19
Bertrand's Postulate, improvements      42
Bertrand's Postulate, prime counting function      22
Bertrand's Postulate, sum of primes      23
Bezout's Lemma      37
Bhaskaracharya      5
Binomial theorem      24 27 63 71 180 263 267
Birch and Swinnerton — Dyer      240
Birch and Swinnerton — Dyer, conjectures      242
Birch and Swinnerton — Dyer, conjectures, congruent numbers      243
Bit length      250
Bit operation      246
Brahmagupta      5
Brun's constant      33
Canonical height      146
Carmichael, number      34 260
Carmichael, number, square-free      261
Carmichael, primitive divisors      169
Carmichael, Theorem      169
Catalan equation      25 57
Cataldi      5 24
Cauchy, formula      173
Cauchy, Residue Theorem      126 131
Cauchy, sequence      147 223
CHARACTER      213 229
Character, Dirichlet      217 227 268
Character, group      214
Character, L-function      217
Character, orthogonality relations      215
Chinese remainder theorem      61 81 256
Chinese Remainder Theorem, proof      62
Class group      90
Class number      81 89
Class number, formula      225
Class number, formula, via quadratic forms      243
Class number, one      243
Clay Mathematics Institute      187 243
Clay Mathematics Institute, Birch and Swinnerton — Dyer Clay Mathematics Institute, conjectures      243
Clay Mathematics Institute, Riemann Hypothesis      187
Clement      33
Cole      27
complexity      247
Complexity, O-notation      247
Complexity, polynomial      250
Congruent number      98 99 103
Congruent number, elliptic curve      100 118
Congruent number, elliptic curve, Zsigmondy's Theorem      116
Congruent number, problem      101
Congruent number, Tunnell's Theorem      100 243
conjugate      86
Convolution      164
Convolution, identity      164
Convolution, inverse      164
Coprime      24 36
Critical strip      185
Cryptography      251
Cryptography, digital signature      252
Cryptography, public, private key      251
Cunningham Project      269 277
Cusp      241
Cyclotomic polynomial      167
De La Vallee Poussin      157
Dedekind, zeta function      229
Dedekind, zeta function, analytic continuation      230
Dedekind, zeta function, Euler product      229
Dedekind, zeta function, residue formula      230
Digital signature      252
Diophantine, approximation      75
Diophantine, equation      43
Diophantine, equation, quadratic      59
Diophantine, equation, solutions modulo p      67
Diophantine, equation, sum of squares      48
Diophantus      5
Dirichlet      5 75 225 230
Dirichlet, character      217 227 268
Dirichlet, character, multiplicative      218
Dirichlet, convolution      164
Dirichlet, convolution, Abelian group      165
Dirichlet, convolution, identity      164
Dirichlet, convolution, inverse      164
Dirichlet, kernel      190
Dirichlet, L-function      217
Dirichlet, L-function, elliptic curve      241
Dirichlet, series      166 180 209 221
Dirichlet, Theorem      211
Discriminant, polynomial      150
Discriminant, quadratic field      92
Discriminant, quadratic form      78
Divisibility sequence      112
Duplication formula      134 141 146
Duplication formula, proof      137
Elementary proof      180
Elliptic curve      54 56 93 127 139
Elliptic curve, $\mathbb{K}$-points      127
Elliptic curve, additive reduction      241
Elliptic curve, affine part      126
Elliptic curve, arithmetic conductor      242
Elliptic curve, associative law      97
Elliptic curve, canonical height      146
Elliptic curve, complex torsion point      129
Elliptic curve, congruent number      100 116
Elliptic curve, cusp      241
Elliptic curve, defined over $\mathbb{k}$      127
Elliptic curve, duplication Elliptic curve, formula      134
Elliptic curve, Fermat's Last Theorem      56
Elliptic curve, forms a group      96
Elliptic curve, generalized Weierstrass equation      109 116
Elliptic curve, Hasse Theorem      240
Elliptic curve, identity      96
Elliptic curve, isogeny      112
Elliptic curve, isomorphism      132
Elliptic curve, L-function      241
Elliptic curve, m-isogenous      113
Elliptic curve, magnified point      113
Elliptic curve, naive height      104
Elliptic curve, non-degeneracy      129 239
Elliptic curve, non-split multiplicative reduction      241
Elliptic curve, other characteristics      109
Elliptic curve, over a finite field      238
Elliptic curve, points of order eleven      110
Elliptic curve, rank      105 240
Elliptic curve, rank, generators      155
Elliptic curve, rational torsion point      129
Elliptic curve, split multiplicative reduction      241
Elliptic curve, torsion point      106 128 155
Elliptic curve, Weierstrass equation      108
Elliptic curve, Weierstrass model      108
Elliptic divisibility sequence      112 118
Elliptic function      125 132
Elliptic function, fundamental domain      125
Elliptic function, periods      125
Eratosthenes      5 16 245
Eratosthenes, sieve      16
Euclid      5 39 41 207
Euclid, infinitely many primes      8
Euclid, Theorem      7
Euclid, Theorem, analytic proof      8
Euclid, Theorem, Goldbach's proof      40
Euclid, Theorem, original proof      39
Euclid, Theorem, sum of prime reciprocals      11
Euclid, Theorem, topological proof      40
Euclidean algorithm      35—37 46 84 138 258 275
Euclidean Algorithm, complexity      254
Euclidean Algorithm, complexity over finite fields      277
Euclidean Algorithm, finite fields      276
Euclidean Algorithm, polynomial      137
Euclidean ring      45 47
Euler      5
Euler, and Mersenne primes      24
Euler, conjecture      57
Euler, conjecture, Elkies      58
Euler, conjecture, Lander and Parkin      57
Euler, Criterion      262
Euler, Fermat Theorem      62
Euler, four-square identity      50
Euler, functional equation      206
Euler, infinitely many primes      8
Euler, Mascheroni constant      10 159 197 199
Euler, phi-function      61 167 218 221 259
Euler, prime values of polynomials      18
Euler, product      14 170
Euler, product, Dedekind zeta function      229
Euler, product, L-function      218
Euler, product, L-function of elliptic curve      241
Euler, product, Riemann zeta function      14
Euler, summation formula      158
Euler, Summation Formula, divisor function      160
Euler, Summation Formula, harmonic series      159
Euler, Summation Formula, Stirling's Formula      160
Extended Riemann hypothesis      268
Factor base method      272
Factorization, $\Gamma$ function      198
Factorization, of a function      2 13 183 198 205
Factorization, Riemann zeta function      15 205
Factorization, sine function      202
Factorizing, factor base method      272
Factorizing, rho method      270
Faltings' Theorem      104 118
Fermat      5
Fermat, and Mersenne primes      24
Fermat, Last Theorem      56 83
Fermat, little theorem      24 34 253 255
Fermat, Little Theorem, base      34
Fermat, Little Theorem, combinatorial proof      24
Fermat, Little Theorem, composite modulus      162
Fermat, Little Theorem, proof using group theory      25
Fermat, numbers      29 40
fibonacci      5 62
Fibonacci, sequence      112
FOURIER      5
Fourier, analysis      42 181 206 234
Fourier, coefficient      189
Fourier, coefficient, periodic function      189
Fourier, series      189
Fourier, transform      188
Fudge      234
Fundamental domain      125
Fundamental Theorem of Arithmetic      7 24 51—54 183 269
Fundamental Theorem of Arithmetic, by inverting primes      55
Fundamental Theorem of Arithmetic, Euclidean rings      47
Fundamental Theorem of Arithmetic, failure      54 73 74
Fundamental Theorem of Arithmetic, Fermat's Last Theorem      83
Fundamental Theorem of Arithmetic, for ideals      84 89
Fundamental Theorem of Arithmetic, for ideals, Dedekind zeta function      229
Fundamental Theorem of Arithmetic, function-theoretic      197 205
Fundamental Theorem of Arithmetic, Gaussian integers      48
Fundamental Theorem of Arithmetic, in other rings      45 47
Fundamental Theorem of Arithmetic, inductive proof      38
Fundamental Theorem of Arithmetic, proof      35
Fundamental unit      85 228
Fundamental unit, class number formula      227
Furstenberg, topological proof of Euclid      40
Galois      5
Galois theory      167
Gamma function      183 197 206
Gamma function, analytic continuation      184
Gamma function, analyticity      184
Gamma function, Stirling's formula      204
Gandhi's prime formula      163
Gauss      5 32 41
Gauss, class number one problem      228
Gauss, function lies in S      188
Gauss, Lemma      153
Gauss, sum      70 233 235
Gaussian integers      45 57
Gaussian integers, and cubic Diophantine equations      52
Gaussian integers, Fundamental Theorem of Arithmetic      48
Gaussian integers, primes      49
gcd( )      3
Generator      112
Goldbach      18 40
Greatest common divisor      36 46
Group of units      46 47 54 62 75
Group theory      25 26 47 60 66 106
hadamard      157
Hardy — Ramanujan number      117
Harmonic analysis      181
Harmonic integral test      10
Harmonic series      9
Hasse, Theorem      109 240 243
Hecke      230
Height      104 113
Height, canonical      146
Height, logarithmic      133
Height, naive      133
Height, projective      133
hilbert      5
Hilbert 10th problem      18
Hilbert, Nullstellensatz      136 155
Homogenous, coordinates      110
Homogenous, polynomial      135
Homogenous, polynomial, defines a morphism      135
Homogenous, polynomial, degree      135
Hurwitz' Lemma      87
Hypatia      5
Ideal      84
Ideal class group      90
Ideal, class group      90
Ideal, class number      89
Ideal, conjugate      86
Ideal, divides      88
Ideal, Fundamental Theorem of Arithmetic      89
Ideal, maximal      89
Ideal, norm      87
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