The purpose of this section is to give an overview of the aims of algebraic number theory and to provide motivation for the study of the subject. We shall be concerned with generalisations of the integral domain ℤ of ordinary integers which are called rings of algebraic integers: an algebraic integer is a root of a monic polynomial in ℤ[X]. Many of the definitions and results of ordinary number theory have natural extensions in algebraic number theory, and, in fact, are often better understood in this wider context. Frequently the study of a suitable ring of algebraic integers will help in the solution of a problem which initially had been stated entirely in terms of ordinary integers: for instance, questions concerning the integral (or rational) solutions of an equation with integral (or rational) coefficients can frequently be dealt with by the study of a suitable ring of algebraic integers. We shall consider a number of instances of this phenomenon.

For ease of exposition we shall introduce a number of concepts in a rather informal manner; they will, of course, all receive a full and formal definition later on. In the same way all results quoted in this introduction will subsequently be proved in the text.

We begin by considering the classical problem of when the prime number p can be represented as the sum of the squares of two integers.