Number Theory Volume I Tools and Diophantine Equations [Paperback]

The central theme of this book is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects. The book contains more than 350 exercises and the text is largely self-contained. Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included five appendices on these techniques.

The central theme of this book is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers.

to Diophantine Equations.- to Diophantine Equations.- Tools.- Abelian Groups, Lattices, and Finite Fields.- Basic Algebraic Number Theory.- p-adic Fields.- Quadratic Forms and Local-Global Principles.- Diophantine Equations.- Some Diophantine Equations.- Elliptic Curves.- Diophantine Aspects of Elliptic Curves.

From the reviews:

The book under review deals with Diophantine analysis from a number-theoretic point of view. & Each chapter ends with exercises, ranging from simple to quite challenging problems. The clarity of the exposition is the one we expect from the author of two highly successful books on computational number theory & and makes this volume a must-read for researchers in Diophantine analysis. (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, October, 2013)

This is the first volume of a highly impressive two-volume textbook on Diophantine analysis. & Readers are presented with an almost overwhelming amount of material. This & text book is bound to become an important reference forl%