ABSTRACT

The notion of valuation is basic in the study of fields. This chapter begins by introducing it, concentrating mainly on the properties needed later, but placing them in a wider context. After defining absolute values, which generalize an idea familiar from real numbers, and exploring the topological and metric consequences, where the triple correspondence: valuations, valuation rings, places is explained and illustrated. The structure of complete fields is further elucidated by the power series representation. The chapter takes a brief look at valuations of higher rank. These results are mainly used to classify the valuations on a function field in two variables, so the reader may wish merely to skim these sections and return to them later, but apart from their importance in the general theory these notions also help to illustrate the special case concerned here. The chapter shows how the localization process can be applied to form the valuation ring of a subordinate valuation.