The compositum of wild extensions of local fields of prime degree

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.