Bringing the material up to date to reflect modern applications, Algebraic Number Theory, Second Edition has been completely rewritten and reorganized to incorporate a new style, methodology, and presentation. This edition focuses on integral domains, ideals, and unique factorization in the first chapter; field extensions in the second chapter; and class groups in the third chapter. Applications are now collected in chapter four and at the end of chapter five, where primality testing is highlighted as an application of the Kronecker–Weber theorem. In chapter five, the sections on ideal decomposition in number fields have been more evenly distributed. The final chapter continues to cover reciprocity laws. New to the Second Edition Reorganization of all chapters More complete and involved treatment of Galois theory A study of binary quadratic forms and a comparison of the ideal and form class groups More comprehensive section on Pollard’s cubic factoring algorithm More detailed explanations of proofs, with less reliance on exercises, to provide a sound understanding of challenging material The book includes mini-biographies of notable mathematicians, convenient cross-referencing, a comprehensive index, and numerous exercises. The appendices present an overview of all the concepts used in the main text, an overview of sequences and series, the Greek alphabet with English transliteration, and a table of Latin phrases and their English equivalents. Suitable for a one-semester course, this accessible, self-contained text offers broad, in-depth coverage of numerous applications. Readers are lead at a measured pace through the topics to enable a clear understanding of the pinnacles of algebraic number theory.
Author(s): Richard A. Mollin
Series: Discrete Mathematics and Its Applications
Edition: 2nd
Publisher: CRC Press
Year: 2011
Language: English
Pages: 417
Tags: Математика;Общая алгебра;
Algebraic Number Theory, Second Edition......Page 2
Discrete Mathematics Its Applications......Page 3
Dedication......Page 6
Contents......Page 7
Preface......Page 9
About The Author......Page 12
Suggested Course Outlines......Page 13
Course Outlines......Page 14
1.1 Integral Domains......Page 15
1.2 Factorization Domains......Page 21
1.3 Ideals......Page 29
1.4 Noetherian and Principal Ideal Domains......Page 34
1.5 Dedekind Domains......Page 39
1.6 Algebraic Numbers and Number Fields......Page 49
1.7 Quadratic Fields......Page 58
2.1 Automorphisms, Fixed Points, and Galois Groups......Page 69
2.2 Norms and Traces......Page 79
2.3 Integral Bases and Discriminants......Page 84
2.4 Norms of Ideals......Page 97
3.1 Binary Quadratic Forms......Page 101
3.2 Forms and Ideals......Page 110
3.3 Geometry of Numbers and the Ideal Class Group......Page 122
3.4 Units in Number Rings......Page 136
3.5 Dirichlet's Unit Theorem......Page 144
4.1 Prime Power Representation......Page 153
4.2 Bachet's Equation......Page 159
4.3 The Fermat Equation......Page 163
4.4 Factoring......Page 179
4.5 The Number Field Sieve......Page 188
5.1 Inertia, Ramification, and Splitting of Prime Ideals......Page 195
5.2 The Different and Discriminant......Page 210
5.3 Ramification......Page 227
5.4 Galois Theory and Decomposition......Page 235
5.5 Kummer Extensions and Class- Field Theory......Page 247
5.6 The Kronecker- Weber Theorem......Page 258
5.7 An Application— Primality Testing......Page 269
6.1 Cubic Reciprocity......Page 275
6.2 The Biquadratic Reciprocity Law......Page 292
6.3 The Stickelberger Relation......Page 308
6.4 The Eisenstein Reciprocity Law......Page 325
Basic Axioms......Page 332
Groups......Page 333
Cosets of Groups......Page 334
Rings and Fields......Page 335
Modules......Page 336
Mappings — Morphisms......Page 339
Rings of Quotients......Page 341
Polynomials and Polynomial Rings......Page 342
Polynomial Congruences......Page 343
Basic Matrix Theory......Page 348
Zorn’s Lemma......Page 352
The Principle of Mathematical Induction......Page 353
The Multinomial Theorem......Page 354
Some Elementary Number Theory......Page 355
Appendix B......Page 358
Power Series for some Elementary Functions......Page 365
The Greek Alphabet......Page 368
Latin Phrases......Page 369
Bibliography......Page 371
Section 1.1......Page 376
Section 1.2......Page 377
Section 1.3......Page 378
Section 1.4......Page 379
Section 1.5......Page 380
Section 1.6......Page 381
Section 1.7......Page 382
Section 2.1......Page 384
Section 2.2......Page 386
Section 2.3......Page 389
Section 2.4......Page 390
Section 3.1......Page 391
Section 3.2......Page 393
Section 3.3......Page 394
Section 3.4......Page 396
Section 3.5......Page 397
Section 4.1......Page 400
Section 4.2......Page 402
Section 4.3......Page 403
Section 4.4......Page 405
Section 5.1......Page 406
Section 5.2......Page 407
Section 5.3......Page 408
Section 5.4......Page 409
Section 5.6......Page 410
Section 6.1......Page 412
Section 6.2......Page 414
Section 6.3......Page 416
Section 6.4......Page 417
