A Classical Introduction to Modern Number Theory

Kenneth Ireland, Michael Rosen (auth.)038797329X, 9780387973296

Bridging the gap between elementary number theory and the systematic study of advanced topics, A CLASSICAL INTRODUCTION TO MODERN NUMBER THEORY is a well-developed and accessible text that requires only a familiarity with basic abstract algebra. Historical developement is stressed throughout, along with wide-ranging coverage of significant results with comparitively elementary proofs, some of them new. An extensive bibliography and many challenging exercises are also included. This second edition has been corrected and contains two new chapters which provide a complete proof of the Mordel-Weil theorem for elliptic curves over the rational numbers, and an overview of recent progress on the arithmetic of elliptic curves.

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Table of contents :
Front Matter….Pages i-xiii
Unique Factorization….Pages 1-16
Applications of Unique Factorization….Pages 17-27
Congruence….Pages 28-38
The Structure of U ( ℤ/nℤ )….Pages 39-49
Quadratic Reciprocity….Pages 50-65
Quadratic Gauss Sums….Pages 66-78
Finite Fields….Pages 79-87
Gauss and Jacobi Sums….Pages 88-107
Cubic and Biquadratic Reciprocity….Pages 108-137
Equations over Finite Fields….Pages 138-150
The Zeta Function….Pages 151-171
Algebraic Number Theory….Pages 172-187
Quadratic and Cyclotomic Fields….Pages 188-202
The Stickelberger Relation and the Eisenstein Reciprocity Law….Pages 203-227
Bernoulli Numbers….Pages 228-248
Dirichlet L -functions….Pages 249-268
Diophantine Equations….Pages 269-296
Elliptic Curves….Pages 297-318
Back Matter….Pages 319-344