# A valuation criterion for normal bases in elementary abelian extensions

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Nigel P. Byott, Jul 16, 2014 Available from:- [Show abstract] [Hide abstract]

**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.Bulletin of the London Mathematical Society 04/2010; DOI:10.1112/blms/bds005 · 0.70 Impact Factor -
##### Article: A Valuation Criterion for Normal Basis Generators of Hopf-Galois Extensions in Characteristic p

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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.05/2009; DOI:10.5802/jtnb.750 - [Show abstract] [Hide abstract]

**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.Journal of Algebra 11/2008; 320(10-320):3811-3820. DOI:10.1016/j.jalgebra.2008.05.024 · 0.60 Impact Factor