Finite Extensions of the Field of p-adic Numbers

Abstract

The field Q p is not algebraically closed: It admits algebraic extensions of arbitrarily large degrees. These extensions are the p-adic fields to be studied here. Each one is a finite-dimensional, hence locally compact, normed space over Q p . A main result is the following: The p-adic absolute value on Q p has a unique extension to any finite algebraic extension K of Q p .