Introduction and Preliminaries

Abstract

Some basic properties of ultrametric fields—their topological structure and geometry are discussed in this chapter. We introduce the p-adic valuation, p being prime, and prove that any valuation of ${{\mathbb {Q}}}$ (the field of rational numbers) is either the trivial valuation, a p-adic valuation or a power of the usual absolute value, where the power is positive and less than or equal to 1. We discuss equivalent valuations too.