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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Cp cross Cp JTNB.pdf - Accepted Version

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 ${\mathfrak{A}}_{H}$.