ABSTRACT
The origin of algebraic number theory is in the ‘attempted-proofs’ of Fermat’s Last Theorem: The motivation is to be found in the generalizations of the integral domainZ (of rational integers) giving rise to the notion of algebraic integers: (See ‘Euclidean domains’, chapter 3). Many of the results of number theory are tackled in a more general set-up in algebraic number theory. For instance, Fermat’s two-square theorem (theorem 4, chapter 1) is proved by considering the Euclidean domain Z[i] of Gaussian integers. We recall that an algebraic integer is a zero of a monic polynomial in Z[x]. Very often, the study of a suitable ring of algebraic integers helps in the solution of a problem which is initially stated in terms of ordinary (rational) integers. For example, we have the context of solutions of the Pell equation x2 − my2 = 1. The consideration of ideals instead of elements of a ring was indeed a breakthrough for purposes of factorization. This is achieved in Dedekind domains considered in chapter 12.
