A valuation criterion for normal bases in elementary abelian extensions
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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 $K$.
In addition to some small notational changes, the title "On the valuation of normal basis generators" was changed. The paper is accepted by the Bulletin of the LMS
In addition to some small notational changes, the title "On the valuation of normal basis generators" was changed. The paper is accepted by the Bulletin of the LMS