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


Authors: Nigel P. Byott and G. Griffith Elder
Journal: Proc. Amer. Math. Soc. 142 (2014), 3705-3712
MSC (2010): Primary 11S15, 11R33
DOI: https://doi.org/10.1090/S0002-9939-2014-12126-5
Published electronically: July 8, 2014
MathSciNet review: 3251712
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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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Additional Information

Nigel P. Byott
Affiliation: College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QE, United Kingdom
Email: N.P.Byott@ex.ac.uk

G. Griffith Elder
Affiliation: Department of Mathematics, University of Nebraska at Omaha, Omaha, Nebraska 68182-0243
Email: elder@unomaha.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12126-5
Keywords: Galois module structure, associated order
Received by editor(s): April 30, 2009
Received by editor(s) in revised form: November 23, 2012
Published electronically: July 8, 2014
Communicated by: Ted Chinburg
Article copyright: © Copyright 2014 American Mathematical Society