The Compositum of Wild Extensions of Local Fields of Prime Degree

Abstract

In this paper we present a general view of the totally and wildly ramified extensions of degree p of a p-adic field K. Our method consists in deducing the properties of the set of all extensions of degree p of K from the study of the compositum ${\cal C}_K(p)$ of all its elements. We show that in fact ${\cal C}_K(p)$ is the maximal abelian extension of exponent p of F = F(K), where F is the compositum of all cyclic extensions of K of degree dividing p − 1. By our method, it is fairly simple to recover the distribution of the extensions of K of degree p (and also of their isomorphism classes) according to their discriminant.