Integral Galois module structure for elementary abelian extensions with a Galois scaffold

Byott, Nigel; Elder, G.

doi:10.1090/S0002-9939-2014-12126-5

Integral Galois module structure for elementary abelian extensions with a Galois scaffold
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by Nigel P. Byott and G. Griffith Elder

Proc. Amer. Math. Soc. 142 (2014), 3705-3712

DOI: https://doi.org/10.1090/S0002-9939-2014-12126-5

Published electronically: July 8, 2014

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Abstract:

This paper justifies an assertion by [Elder, Proc. Amer. Math. Soc., 2009] that Galois scaffolds make the questions of Galois module structure tractable. Let $k$ be a perfect field of characteristic $p$ and let $K=k((T))$. For the class of characteristic $p$ elementary abelian $p$-extensions $L/K$ with Galois scaffolds described in loc. cit., we give a necessary and sufficient condition for the valuation ring $\mathfrak {O}_L$ to be free over its associated order $\mathfrak {A}_{L/K}$ in $K[\mathrm {Gal}(L/K)]$. Interestingly, this condition agrees with the condition found by Y. Miyata concerning a class of cyclic Kummer extensions in characteristic zero.

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