p-adic Numbers

Abstract

Having built our foundation, we can now apply the general theory to the specific case of the field ℚ of rational numbers. Extending our scope to include all fields of algebraic numbers (i.e., finite extensions of ℚ), or even to include what the experts call “global fields” in general, would not be very hard. Nevertheless, we have preferred to stick, at first, to the most concrete example available. In a later chapter, we will consider some aspects of the problem of extending valuations from ℚ to larger fields. More details about the theory of valuations on global fields can be found in several of the references.