Valuated Fields

The descriptive language of general topology is known to all mathematicians. The concept of a valuation allows one to introduce this language into the theory of algebraic numbers in a natural and fruitful way. We therefore propose to study some of the connections between valuation theory, algebraic number theory, and topology. Strictly speaking the topological considerations are just of a conceptual nature and in fact only the most elementary results on metric spaces and topological groups will be used; nevertheless these considerations are essential to the point of view taken throughout this chapter and indeed throughout the entire book.