INTRODUCTION

In this Chapter we study the complete valued fields which will arise when we look at algebraic number fields in Chapter 10. The first two sections are straightforward applications and amplifications of the results of the previous Chapter and prepare the way for Chapter 10. The remainder of the Chapter gives a couple of results which are important in the further development of the theory but which are not required further in this book.

We start by defining the fields we shall be considering:

DEFINITION 1.1. Let the field k be complete with respect to the (nonarch.) valuation | |. We say that k is a p-adic field if

1. (i) k has characteristic 0

2. (ii) | | is discrete

3. (iii) the residue class field ρ is finite.

We can give at once an alternative characterization:

LEMMA 1.1. The valued field k is a p-adic field if and only if it is a finite extension of ℚpfor some p.

Proof. (i) Suppose that k is a finite extension of ℚp. Then it is a p-adic field by Lemma 4.1 and 5.1 of Chapter 7.

(ii) Let k be a p-adic field. Then k ⊃ ℚ by (i) of the definition. Since the residue class field is finite by (iii) of the definition, it has characteristic p for some prime p. Hence the valuation on k induces a valuation equivalent to the p-adic valuation.