In this chapter we seek to lay down the algebraic foundations which will be needed. In the first section we recall a number of basic results from field theory, and we then briefly consider the theory of finite commutative algebras over a field. In the second section we introduce the notion of integrality and the Noetherian properties for modules and rings.

The reader is advised not to spend too much time on, Chapter I, and to move to Chapter II - where the algebraic number theory really begins. Indeed, for the reader who has already encountered these four topics – in some form or other – it is suggested that he start straightaway at Chapter II; and then refer back to Chapter I as is necessary.

Fields and Algebras

As has been seen in the introduction, many arithmetic problems require us to work with fields. First and foremost among the types of field which we need to consider are algebraic number fields, that is to say extensions of finite degree over the rationals. Reduction techniques lead us to work with finite fields, which turn up as the residue class fields of rings of algebraic integers modulo prime ideals. Finally we shall have to consider various “completions”, of which the fields of real or of complex numbers are the most familiar examples.

We shall freely assume basic field theory, including Galois theory, but we include here a treatment of certain special topics, which are important m-th roots of unity, can be written as a product

where the product extends over certain residue class characters φ, including φ = ε.