A Classical Introduction to Modern Number Theory
2nd Edition
Published in October 1990
Book
Hardback
XIV, 389 pages
978-3-540-97329-4 (ISBN)
Description
Bridging the gap between elementary number theory and the systematic study of advanced topics. This text requires a familiarity with basic abstract algebra. Historical development is stressed throughout, along with coverage of significant results with comparatively elementary proofs, some of them new. An bibliography and several exercises are also included and this edition offers two new chapters that give a complete proof of Mordell's fundamental theorem and an overview of Flating's proof of the Mordell conjecture. Also included is material on the recent progress on the arithmetic of elliptic curves.
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Content
Contents: Unique Factorization.- Applications of Unique Factorization.- Congruence.- The Structure of U(Z/nZ).- Quadratic Reciprocity.- Quadratic Gauss Sums.- Finite Fields.- Gauss and Jacobi Sums.- Cubic and Biquadratic Reciprocity.- Equations Over Finite Fields.- The Zeta Function.- Algebraic Number Theory.- Quadratic and Cyclotomic Fields.- The Stickelberger Relation and the Eisenstein Reciprocity Law.- Bernoulli Numbers.- Dirichlet L-Functions.- Diophantine Equations.- Elliptic Curves.- The Mordell-Weil Theorem.- New Progress in Arithmetic Geometry.- Selected Hints for the Exercises.