Integral Galois Module Structure for Elementary Abelian Extensions with a Galois Scaffold

Details Integral Galois Module Structure for Elementary Abelian Extensions with a Galois Scaffold Integral Galois Module Structure for Elementary Abelian Extensions with a Galois Scaffold

2009-08-31

arxiv.org/abs/0908.4562v1

Mathematics

Number Theory

Scientific paper

This paper justifies an assertion in (Elder, Proc AMS 137 (2009), no 4, 1193--1203) 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 mentioned paper, we give a necessary and sufficient condition for the valuation ring $\mathfrak{O}_L$ to be free over its associated order $\mathcal{A}_{L/K}$ in $K[\Gal(L/K)]$. 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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