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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.
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
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