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(5). DOI: 10.1112/blms/bdm036
Source: arXiv


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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Available from: Nigel P. Byott, Jul 16, 2014
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    • "For cyclic extensions of degree p this condition is also sufficient; cf. [2]. However, we will see in Example 4.1 that this condition is not sufficient for cyclic extensions of degree p 2 . "
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    ABSTRACT: If $L/K$ is a finite Galois extension of local fields, we say that the valuation criterion $VC(L/K)$ holds if there is an integer $d$ such that every element $x \in L$ with valuation $d$ generates a normal basis for $L/K$. Answering a question of Byott and Elder, we first prove that $VC(L/K)$ holds if and only if the tamely ramified part of the extension $L/K$ is trivial and every non-zero $K[G]$-submodule of $L$ contains a unit. Moreover, the integer $d$ can take one value modulo $[L:K]$ only, namely $-d_{L/K}-1$, where $d_{L/K}$ is the valuation of the different of $L/K$. When $K$ has positive characteristic, we thus recover a recent result of Elder and Thomas, proving that $VC(L/K)$ is valid for all extensions $L/K$ in this context. When $\char{\;K}=0$, we identify all abelian extensions $L/K$ for which $VC(L/K)$ is true, using algebraic arguments. These extensions are determined by the behaviour of their cyclic Kummer subextensions.
    Full-text · Article · Apr 2010 · Bulletin of the London Mathematical Society
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    • "In the (characteristic 0) situation where K a finite extension of the field Q p of p-adic numbers, the author and Elder [2] showed the existence of integer certificates for normal basis generators in totally ramified elementary abelian extensions L/K, under the assumption that L/K contains no maximally ramified subfield. This assumption is necessary, since there can be no integer certificate in the case L = K( p √ π) with v K (π) = 1. "
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    ABSTRACT: Let S/R be a finite extension of discrete valuation rings of characteristic p>0, and suppose that the corresponding extension L/K of fields of fractions is separable and is H-Galois for some K-Hopf algebra H. Let D_{S/R} be the different of S/R. We show that if S/R is totally ramified and its degree n is a power of p, then any element $\rho$ of L with $v_L(\rho)$ congruent to $-v_L(D_{S/R})-1$ mod n generates L as an H-module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. Elder for Galois extensions.
    Full-text · Article · May 2009 · Journal de Theorie des Nombres de Bordeaux
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    • "This is due to Artin–Schreier theory and the fact that the residue field is perfect . Precisely, in their theorem, Byott and Elder's hypothesis is a restriction to extensions that do not contain any totally ramified Kummer extension given by a pth root of some uniformizing element of the ground field: the ramification jump of such extension is divisible by p and no valuation criterion for normal basis generators is satisfied (see Example 1 of [1]). In characteristic p, this obstacle does not appear for totally ramified abelian p-extensions because, for such extensions, all lower ramification jumps are relatively prime to p, whenever the residue field is perfect [7]. "
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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.
    Full-text · Article · Nov 2008 · Journal of Algebra
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