Keele Research Repository

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} $.