A Classical Introduction to Modern Number Theory

Produktbeschreibung

Bridging the gap between elementary number theory and the systematic study of advanced topics, BA Classical/B BIntroduction to Modern Number Theory/B 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. TOC: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.- Bibliography.- Index. 

Inhaltsverzeichnis

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. 

Kritik

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. Gouvˆa, MathDL, January, 2006) 

Autoreninfo

Michael Rosen is a hugely bestselling author of poetry. Michael frequently appears on radio and gives talks. He is the Children's Laureate for 2007-2009 and the winner of the Eleanor Farjeon Award for services to children's literature. Michael lives in London.