Truman, PJ (2016) Hopf-Galois module structure of tame biquadratic extensions. Journal de Theorie des Nombres de Bordeaux, 28 (2). pp. 557-582.

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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
Uncontrolled Keywords: Hopf-Galois Theory; Galois module structure; tame extension
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Natural Sciences > School of Computing and Mathematics
Depositing User: Symplectic
Date Deposited: 04 Jan 2017 15:24
Last Modified: 28 Feb 2020 10:54

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