A valuation criterion for normal bases in elementary abelian extensions

Abstract

Let $p$ be a prime number and let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers. Let $N$ be a fully ramified, elementary abelian extension of $K$. Under a mild hypothesis on the extension $N/K$, we show that every element of $N$ with valuation congruent mod $[N:K]$ to the largest lower ramification number of $N/K$ generates a normal basis for $N$ over $K$.