This well-developed, accessible text details the historical development of the subject throughout. It also provides wide-ranging coverage of significant results with comparatively elementary proofs, some of them new. This second edition contains two new chapters that 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.

1.unique factorization
2.applications of unique factorization
3.congruence
4.the stucture of U
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 ralation 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

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