Edwards H.M. — Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory :: Электронная библиотека попечительского совета мехмата МГУ

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Edwards H.M. — Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory

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Название: Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory

Автор: Edwards H.M.

Аннотация:

This book is an introduction to algebraic number theory via the famous problem of "Fermat's Last Theorem." The exposition follows the historical development of the problem, beginning with the work of Fermat and ending with Kummer's theory of "ideal" factorization, by means of which the theorem is proved for all prime exponents less than 37. The more elementary topics, such as Euler's proof of the impossibilty of x+y=z, are treated in an elementary way, and new concepts and techniques are introduced only after having been motivated by specific problems. The book also covers in detail the application of Kummer's ideal theory to quadratic integers and relates this theory to Gauss' theory of binary quadratic forms, an interesting and important connection that is not explored in any other book.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1st edition

Год издания: 1996

Количество страниц: 410

Добавлена в каталог: 04.04.2008

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Предметный указатель

$\sigma$ (conjugation of cyclotomic integers)      108
Ambiguous divisor classes      298—301
Analytic method      5 89 92 113 246
Bachet      2
Bachet, commentary on Diophantus      11
Bernoulli numbers      230—231
Bernoulli polynomials      229
Bhascara Acharya      28 30n
Binary quadratic forms      314
Brahmagupta      27
Brouncker      30—32
Cauchy      77—79
Character of a divisor      294
Character of a divisor, of a divisor class      294 334
Characters mod a prime      188
Characters mod a prime, mod an integer      366
Chinese remainder theorem      375
Chinese remainder theorem, for divisors      147
Circulatory matrices      207
Class field theory      288n
Class number      163
Class number formula      225 346 349 351 364
Class number, second factor of      210 214 225
Composition of forms      334—338
Computation, attitudes toward      81 226n
Congruence of natural numbers      374
Congruence, mod a cyclotomic integer      85 90
Congruence, mod a divisor      143
Congruence, mod a prime divisor      127
Conjugates of a cyclotomic integer      83
Conjugates of a cyclotomic integer, of a quadratic integer      246
Continued fractions      278n 313
Convenient numbers      298 310—312
Cyclic method      27—30 265
Cyclotomic integers      81
Dedekind      144
Dickson      13
Diophantus      2 26
Dirichlet      65 74
Division of cyclotomic integers      85
Divisor class group for an order of quadratic integers      309
Divisor, of cyclotomic integers      139
Divisor, of quadratic integers      256 258
Equivalence of binary quadratic forms      315
Equivalence of divisors      162 256 309
Equivalence of units      213
Euclid's formula for perfect numbers      19
Euclidean algorithm      373—374
Euler product formula      182
Euler product formula, for cyclotomic integers      183 184—185
Euler product formula, for quadratic integers      342
Fermat      1—2
Fermat numbers      23—25
Fermat's Last Theorem, origin of name      2
Fermat's Last Theorem, statement      2
Fermat's theorem      23
Fermat, challenge to English      25—26
Fermat, discoveries in number theory      36—38
Finiteness of class number      163—165 261
Fourier analysis (finite dimensional)      207
Fundamental Theorem of Arithmetic      376
Fundamental theorem of divisor theory      137

Fundamental unit of an order of quadratic integers      271
Gaussian sums      360
Genus of a divisor class      296 334
Germain, Sophie      61
Germain, theorem of      64
Girard's theorem      15—16
Ideal      144
Ideal prime factors      see “Prime divisors”
Idoneal numbers      see “Convenient numbers”
Index of an order of quadratic integers      308
Infinite descent, method of      8—9 373
Irreducible cyclotomic integers      84
Jacobi symbol      290
kronecker      86 174 291
Kummer's lemma      243
L-functions      188 352 366—371
L-functions, values at s = 1      195—196 352—361
Lagrange      32 51 60
Lagrange, counterexample to Euler's conjecture      298
lame      73 76—79
legendre      60 65 70
Legendre symbol      177 290n
Leibniz's formula      346 356 358
Liouville      77
Mersenne primes      20
Multiplicities      135 140
Norm, of a cyclotomic integer      83
Norm, of a divisor      145—146 257—258
Norm, of a quadratic integer      246
Orders of quadratic integers      308
Pell      33
Periods $\eta$       108
Prime cyclotomic integers      84
Prime divisors      106 126—127 136—137 246—252
Primitive binary quadratic forms      321
Primitive roots      378—379
Principal      330—331
Proper equivalence of binary quadratic forms      319—320 336—337
Proper representations      317n
Properly primitive binary quadratic forms      321
Quadratic integers      251
Quadratic reciprocity, equivalence to Euler's theorems      290—292
Quadratic reciprocity, priority of discovery      287n
Quadratic reciprocity, proof      175—177 303—304
Quadratic reciprocity, statement      176 178—179 290n
Quadratic reciprocity, supplementary laws      178
Ramifications of primes      252
Reduced divisors      277
Regular primes      168
Regular primes, criterion for      244
Representations of numbers by forms      314—317
Representative set      163 272
Riemann zeta function      185
Second factor of the class number      see “Class number”
Splitting classes      288
Splitting of primes      252
Summation by parts      190 192
Supplementary laws      see “Quadratic reciprocity”
Unique factorization into primes      77 376
Unique factorization into primes, failure      103
UNIT      84
Wallis      30—33
Zeta function      185

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