A Classical Introduction to Modern Number Theory

by Kenneth Ireland, Michael Rosen

Overview

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 development is stressed throughout, along with wide-ranging coverage of significant results with comparatively 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 Mordell-Weil theorem for elliptic curves over the rational numbers, and an overview of recent progress on the arithmetic of elliptic curves.

Product Details

ISBN-13:	9781441930941
Publisher:	Springer New York
Publication date:	12/01/2010
Series:	Graduate Texts in Mathematics , #84
Edition description:	Second Edition 1990
Pages:	394
Product dimensions:	6.10(w) x 9.25(h) x (d)

Table of Contents

1 Unique Factorization.- 2 Applications of Unique Factorization.- 3 Congruence.- 4 The Structure of U(?/n?).- 5 Quadratic Reciprocity.- 6 Quadratic Gauss Sums.- 7 Finite Fields.- 8 Gauss and Jacobi Sums.- 9 Cubic and Biquadratic Reciprocity.- 10 Equations over Finite Fields.- 11 The Zeta Function.- 12 Algebraic Number Theory.- 13 Quadratic and Cyclotomic Fields.- 14 The Stickelberger Relation and the Eisenstein Reciprocity Law.- 15 Bernoulli Numbers.- 16 Dirichlet L-functions.- 17 Diophantine Equations.- 18 Elliptic Curves.- 19 The Mordell-Weil Theorem.- 20 New Progress in Arithmetic Geometry.- Selected Hints for the Exercises.

Editorial Reviews

From the reviews of the second edition:

K. Ireland and M. Rosen

A Classical Introduction to Modern Number Theory

"Many mathematicians of this generation have reached the frontiers of research without having a good sense of the history of their subject. In number theory this historical ignorance is being alleviated by a number of fine recent books. This work stands among them as a unique and valuable contribution."

— MATHEMATICAL REVIEWS

"This is a great book, one that does exactly what it proposes to do, and does it well. For me, this is the go-to book whenever a student wants to do an advanced independent study project in number theory. … for a student who wants to get started on the subject and has taken a basic course on elementary number theory and the standard abstract algebra course, this is perfect." (Fernando Q. Gouvêa, MathDL, January, 2006)

From the Publisher

To the edition of 1980 have been added two new chapters, intended to convey the essence of recent developments in this ancient but ever- lively field. The text--addressed to graduate students and to advanced undergraduates--treats all the issues standard to its subject matter, but concentrates when possible on those aspects of the field which are at present "research active", and generally avoids complicated analytical machinery. Respectful attention is given to the history of the subject, and there are exercises (with hints) at the ends of the twenty chapters. Superbly produced. (NW) Annotation c. Book News, Inc., Portland, OR (booknews.com)

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