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Reid L.W. — The Elements of the Theory of Algebraic Numbers

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Название: The Elements of the Theory of Algebraic Numbers

Автор: Reid L.W.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1910

Количество страниц: 454

Добавлена в каталог: 15.04.2006

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Предметный указатель

0-function, expression for power of prime ideal      359
0-function, for ideals, definition      358
0-function, general expression      38 44 53 359-362 366 367
0-function, in k( $\mathfrak{i}$ )      185-188
0-function, in R, definition      37
0-function, of higher order      54 367
0-function, product theorem      45 360 361
0-function, summation theorem      46 75 362 363 367
Ambiguous ideal      347
Appertains, exponent to which an integer      99 393
Associated integers, in k( $\mathfrak{i}$ )      163
Associated integers, in k( $\sqrt{-3}$ )      223
Associated integers, in k( $\sqrt{-5}$ )      246
Associated integers, in R      9;
Basis of conjugate ideals      301
Basis of ideals      293-295
Basis, of ideal      293-295
Basis, of ideal, determination      351-355
Basis, of k( $\mathfrak{i}$ )      159-161
Basis, of k( $\sqrt{-3}$ ))      220
Basis, of k( $\sqrt{2}$ )      232
Basis, of k( $\sqrt{5}$ )      245
Basis, of k( $\sqrt{m}$ )      284-287
Basis, of k( $\sqrt{m}$ ), determination      289—292
Binomial congruences      110-112
Biquadratic residues and reciprocity law      205-217
Canonical basis of ideals      294
Character of an integer, biquadratic      209 212
Character of an integer, quadratic, in k( $\mathfrak{i}$ )      212
Character of an integer, quadratic, in R      121
Class number of a realm is finite      437
Class number of a realm, definition      434
Class number of a realm, determination      437-448 451
Classes, ideal, definition      432
Classification of the numbers of an ideal with respect to another ideal      326-330
Common divisor of ideals      303
Congruences in k( $\mathfrak{i}$ )      180
Congruences of condition      59-61 190 369-372
Congruences of first degree in one unknown      68-70
Congruences of nth degree in one unknown, preliminary discussion      66-68 374 375
Congruences of second degree with one unknown      119-121
Congruences of two polynomials      57 370
Congruences, $x^{n}\equiv b, mod p$       114-116
Congruences, $x^{\varphi(m)}$ -1 $\equiv o, mod m$       90 387 388
Congruences, common roots      92 93 389
Congruences, composite modulus      95-97 391 392
Congruences, definition      31 297 323
Congruences, determination      93 94 386
Congruences, elementary theorems      32-37 323-326
Congruences, equivalent systems      64;
Congruences, Euler’s criterion      115
Congruences, limit to number of roots      89 386
Congruences, multiple roots, definition      89 386
Congruences, primitive and imprimitive roots      111
Congruences, root      66 374
Congruences, solution of $x^{1}\equiv -1, mod p, by means of Wilson’s theorem      129 130
Congruences, transformations      62-64 372 374
Congruences, with prime modulus      88-90 385-387
Conjugate ideals      301
Conjugate, numbers      4
Conjugate, realm      4
Dirichlet’s theorem regarding infinity of primes in an arithmetical progression      11
Discriminant, of k( $\mathfrak{i}$ )      161
Discriminant, of k( $\sqrt{-3}$ )      221
Discriminant, of k( $\sqrt{-5}$ )      245
Discriminant, of k( $\sqrt{2}$ )      232
Discriminant, of k( $\sqrt{m}$ )      287 288
Discriminant, of number      284
Divisibility of ideals      263 303
Divisor, greatest common, discussion of definition      252
Divisor, greatest common, in k( $\mathfrak{i}$ )      173
Divisor, greatest common, in R      16 18 25
Divisor, greatest common, of two ideals      310-313 318
Divisors, of integers in R, number of      23
Divisors, of integers in R, sum of      24;
Divisors, of of ideal, number of      318
Equality of ideals      258 259 302
Equivalence of ideals      427-431
Equivalence of ideals in narrower sense      431
Equivalent congruences      62-64 372 373
Eratosthenes, sieve of      10
Euler's criterion for solvability of $x^{n}\equiv b, mod p$       115 122
Factorization of a rational prime determined by (d/p), in k( $\mathfrak{i}$ )      179
Factorization of a rational prime determined by (d/p), in k( $\sqrt{-3}$ )      229
Factorization of a rational prime determined by (d/p), in k( $\sqrt{m}$ )      347 348
Fermat’s theorem      57
Fermat’s theorem as generalized by Euler      57
Fermat’s theorem, analogue for ideals      368 369
Fermat’s theorem, analogue for k( $\mathfrak{i}$ )      189
Frequency of the rational primes      11
Galois realm      281
Gauss’ Lemma      130
General algebraic integers      1 275-279
Generation of realm      3
Ideal numbers, nature explained      254-257
Ideal numbers, necessity for      253
Ideals, definition      257 293
Ideals, determination of basis      298—301
Ideals, introduction of numbers into and omission from symbol      258 295 296
Ideals, numbers defining      295
Ideals, principal and non-principal      260 261 297
Imprimitive numbers      see primitive numbers
Incongruent numbers, complete system of, in k( $\mathfrak{i}$ )      182-185
Incongruent numbers, complete system of, in k( $\sqrt{m}$ )      326
Incongruent numbers, complete system of, in R      34
Index, of a power      106 399
Index, of a product      106 399
Indices, definition      105 399
Indices, solution of congruences by means of      108—110 400-402
Indices, system of      106 399
Integers, absolute value in R      7 33
Integers, of k( $\mathfrak{i}$ )      157
Integers, of k( $\sqrt{-3}$ )      219
Integers, of k( $\sqrt{-5}$ )      245
Integers, of k( $\sqrt{2}$ )      231
Integers, of k( $\sqrt{m}$ )      284-287
Integers, of R      7 23
Legendre’s symbol      127
Multiple, least common, in R      25
Multiple, least common, of two ideals      310-312 318
Multiplication of ideals      261 262 302 303
Non-equivalent ideals, complete system of      434
Norm of ideals      326-338 351
Norm, of a number, determination      351
Norm, of a number, in k( $\mathfrak{i}$ )      156
Norm, of a number, in k( $\sqrt{-3}$ )      218 221

Norm, of a number, in k( $\sqrt{-5}$ )      245
Norm, of a number, in k( $\sqrt{2}$ )      231
Norm, of a number, in k( $\sqrt{m}$ )      283
Norm, of a number, of an ideal, definition      326 337
Norm, of a number, value      330
Norm, of a prime ideal      338
Norm, of a principal ideal      337
Norm, of a product of ideals      334
Number class, ideal modulus      324
Number class, rational modulus      32 33
Numbers of ideals      293
Numbers, algebraic, conjugate      4
Numbers, algebraic, definition      1
Numbers, algebraic, degree of      1
Numbers, algebraic, of k( $\mathfrak{i}$ )      155
Numbers, algebraic, of k( $\sqrt{-3}$ )      1 218
Numbers, algebraic, of k( $\sqrt{-5}$ )      245
Numbers, algebraic, of k( $\sqrt{2}$ )      231
Numbers, algebraic, of k( $\sqrt{m}$ )      281
Numbers, algebraic, of R      7
Numbers, algebraic, of the general realm      271-279
Numbers, algebraic, rational equation of lowest degree satisfied by      2 273
Pell’s equation      423-426
Polynomials in a single variable      268-271
Polynomials with respect to a prime modulus, reduced      62
Polynomials, associated      77 381
Polynomials, common divisor of      76 380
Polynomials, common multiple of      76 380
Polynomials, congruence with respect to a double modulus      81
Polynomials, degree of      76
Polynomials, determination of prime      78 381 382
Polynomials, divisibility of      76 380
Polynomials, division of one by another      382
Polynomials, primary      78 381
Polynomials, prime      78 381
Polynomials, unique factorization theorem for      82-87 382-385
Polynomials, unit      77 381
Power of a prime by which $\mathfrak{m}!$ is divisible      26
Primary integers of k( $\mathfrak{i}$ )      193-196
Prime factors, resolution of an ideal into      348-350
Prime ideals      263-265 304
Prime ideals, determination and classification      339-348
Prime ideals, of k( $\sqrt{-5}$ )      263-265
Prime ideals, of k( $\sqrt{m}$ ), definition      304
Prime numbers, infinite in number      10
Prime numbers, of k( $\mathfrak{i}$ ), classification      177
Prime numbers, of k( $\mathfrak{i}$ ), definition      165
Prime numbers, of k( $\sqrt{-3}$ ), classification      227-230
Prime numbers, of k( $\sqrt{-3}$ ), definition      223
Prime numbers, of k( $\sqrt{-5}$ )      246 247
Prime numbers, of k( $\sqrt{2}$ ), classification      238-240
Prime numbers, of k( $\sqrt{2}$ ), definition      235
Prime numbers, of R, definition      9
Primitive numbers, of k( $\mathfrak{i}$ )      157
Primitive numbers, of k( $\sqrt{-3}$ )      218
Primitive numbers, of k( $\sqrt{m}$ )      282 283
Primitive numbers, of the general realm      274 275;
Primitive numbers, with respect to a prime ideal modulus      398
Primitive root, definition      100
Primitive root, determination      112
Primitive root, of prime of form $2^{2}+1$       151
Primitive root, of prime of form is 2      152
Principal class      432
Product of classes      432
Realm, conjugate      4
Realm, definition      3
Realm, degree      4
Realm, generation      3
Realm, number defining      4 280
Realm, number generating      4
Reciprocal classes      434
Reciprocity law, determination of value of (a/p) by means of      144
Reciprocity law, for biquadratic residues      210 215-217
Reciprocity law, for quadratic residues, in k( $\mathfrak{i}$ )      201-205
Reciprocity law, for quadratic residues, in R      135
Reciprocity law, other applications of      149
Residue system, complete, in k( $\mathfrak{i}$ )      182-185
Residue system, complete, in R      33 34
Residue system, complete,in k( $\sqrt{m}$ )      326
Residue system, reduced, in k( $\mathfrak{i}$ )      185
Residue system, reduced, in k( $\sqrt{m}$ )      358
Residue system, reduced, in R      37
Residue, odd prime moduli of which an integer is a quadratic      128 145 147
Residue, prime moduli of which 2 is a quadratic      133
Residue, prime moduli of which — 1, is a quadratic      128
Residues of powers, complete system of      98 393
Residues of powers, definition      98 392
Residues of powers, law of periodicity      100
Residues, biquadratic      205-217
Residues, cubic      250
Residues, determination of quadratic      124
Residues, n-ic      116
Residues, quadratic non-      121
Residues, quadratic, in k( $\mathfrak{i}$ )      196-201
Residues, quadratic, in R      121
Residues, with respect to a series of moduli, integer having certain      70
Rummer’s ideal numbers      267
Sub-realm      157
Symbol of ideals      257 295
Symbol, Legendre’s      127
Symbol, Legendre’s for ideal      257 295
Unique factorization theorem, for ideals in k( $\sqrt{m}$ )      305-317
Unique factorization theorem, graphical discussion of      169
Unique factorization theorem, in k( $\mathfrak{i}$ )      167 174
Unique factorization theorem, in k( $\sqrt{-3}$ )      226
Unique factorization theorem, in k( $\sqrt{-5}$ ), failure of      247-253
Unique factorization theorem, in k( $\sqrt{2}$ )      236 237;
Unique factorization theorem, in R      12
Unique factorization theorem, necessity for      253
Unique factorization theorem, realms in which original method of proof holds      248-250
Unique factorization theorem, restoration in terms of ideal factors      265 266
Unit ideal, of k( $\sqrt{m}$ )      304
Unit ideal, of k( $\sqrt{z}$ )      263
Unit, fundamental, determination      420-426
Unit, fundamental, of k( $\sqrt{2}$ )      233
Unit, fundamental, of k( $\sqrt{m}$ ), definition      420
Units, of k( $\mathfrak{i}$ )      163
Units, of k( $\sqrt{-3}$ )      222
Units, of k( $\sqrt{-5}$ )      246
Units, of k( $\sqrt{2}$ )      232-235
Units, of k( $\sqrt{m}$ ), definition      403
Units, of R      8
Units, realm imaginary      404
Units, realm real      405-426
Wilson’s Theorem      91
Wilson’s theorem analogue for ideals      388 389
Wilson’s theorem as generalized by Gauss      91

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