Truman, PJ (2016) Hopf-Galois module structure of tame Cp×Cp extensions. Journal de Theorie des Nombres de Bordeaux, 28 (2). 557 - 582. ISSN 1246-7405

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Abstract

Let $ p $ be an odd prime number, $ K $ a number field containing a primitive $ p^{th} $ root of unity, and $ L $ a Galois extension of $ K $ with Galois group isomorphic to $ C_{p} \times C_{p} $. We study in detail the local and global structure of the ring of integers $ {\mathfrak{O}}_{L} $ as a module over its associated order $ {\mathfrak{A}}_{H} $ in each of the Hopf algebras $ H $ giving nonclassical Hopf-Galois structures on the extension, complementing the $ p=2 $ case considered in [12]. For each Hopf algebra giving a nonclassical Hopf-Galois structure on $ L/K $ we show that $ {\mathfrak{O}}_{L} $ is locally free over its associated order $ {\mathfrak{A}}_{H} $ in $ H $, compute local generators, and determine necessary and sufficient conditions for $ {\mathfrak{O}}_{L} $ to be free over $ {\mathfrak{A}}_{H} $.

Item Type: Article
Additional Information: This work is made available online in accordance with the publishers policies. This is the author created, accepted version manuscript following peer review, and may differ slightly from the final published version.
Uncontrolled Keywords: Hopf-Galois; Galois module structure; tame extension
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Natural Sciences > School of Computing and Maths
Depositing User: Symplectic
Date Deposited: 22 Dec 2016 11:08
Last Modified: 01 Jan 2017 01:30
URI: http://eprints.keele.ac.uk/id/eprint/2697

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