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Hopf-Galois module structure of tame biquadratic extensions

Truman

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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} $.

Acceptance Date Sep 5, 2014
Publication Date Jan 1, 2016
Publicly Available Date Mar 29, 2024
Journal Journal de Theorie des Nombres de Bordeaux
Print ISSN 1246-7405
Pages 557-582
DOI https://doi.org/10.5802/jtnb.953
Keywords Hopf-Galois Theory; Galois module structure; tame extension
Publisher URL https://doi.org/10.5802/jtnb.953

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