Bridging the gap between elementary number theory and the systematic study of advanced topics, you are 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 comparitively elementary proofs, some of them new. An extensive bibliography and many challenging exercises are also included. You have been corrected and contain 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.