# Hopf-Galois Module Structure Of Some Tamely Ramified Extensions

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 Title: Hopf-Galois Module Structure Of Some Tamely Ramified Extensions Author: Truman, Paul James Advisor: Byott, Nigel Publisher: University of Exeter Date Issued: 2009-01-12 URI: http://hdl.handle.net/10036/71817 Abstract: We study the Hopf-Galois module structure of algebraic integers in some finite extensions of $p$-adic fields and number fields which are at most tamely ramified. We show that if $L/K$ is a finite unramified extension of $p$-adic fields which is Hopf-Galois for some Hopf algebra $H$ then the ring of algebraic integers $\OL$ is a free module of rank one over the associated order $\AH$. If $H$ is a commutative Hopf algebra, we show that this conclusion remains valid in finite ramified extensions of $p$-adic fields if $p$ does not divide the degree of the extension. We prove analogous results for finite abelian Galois extensions of number fields, in particular showing that if $L/K$ is a finite abelian domestic extension which is Hopf-Galois for some commutative Hopf algebra $H$ then $\OL$ is locally free over $\AH$. We study in greater detail tamely ramified Galois extensions of number fields with Galois group isomorphic to $C_{p} \times C_{p}$, where $p$ is a prime number. Byott has enumerated and described all the Hopf-Galois structures admitted by such an extension. We apply the results above to show that $\OL$ is locally free over $\AH$ in all of the Hopf-Galois structures, and derive necessary and sufficient conditions for $\OL$ to be globally free over $\AH$ in each of the Hopf-Galois structures. In the case $p = 2$ we consider the implications of taking $K = \Q$. In the case that $p$ is an odd prime we compare the structure of $\OL$ as a module over $\AH$ in the various Hopf-Galois structures. Type: Thesis or dissertation Keywords: Hopf-Galois StructureHopf AlgebraHopf OrderGalois Module StructureAlgebraic Number Theory Funders/Sponsor: Engineering And Physical Sciences Research Council

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