A valuation criterion for normal bases in elementary abelian extensions

University of Exeter, Exeter, England, United Kingdom
Bulletin of the London Mathematical Society (Impact Factor: 0.7). 05/2006; 39. DOI: 10.1112/blms/bdm036
Source: arXiv

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 $K$.

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    ABSTRACT: We answer a recent conjecture of [N.P. Byott, G.G. Elder, A valuation criterion for normal bases in elementary abelian extensions, Bull. London Math. Soc. 39 (5) (2007) 705–708] in a more general setting. Precisely, let L/K be a finite abelian p-extension of local fields of characteristic p>0 that is totally ramified. Let b denote the largest ramification break in the lower numbering. We prove that any element x∈L whose valuation over L is equal to b modulo [L:K] generates a normal basis of L/K. The arguments will develop certain properties of ramification groups and jumps, as well as the algebraic structure of certain group algebras.
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