Hopf-Galois Module Structure Of Some Tamely Ramified Extensions

 dc.contributor.author Truman, Paul James en_GB dc.date.accessioned 2009-06-29T10:31:10Z en_GB dc.date.accessioned 2011-01-25T16:55:22Z en_US dc.date.accessioned 2013-03-21T12:04:19Z dc.date.issued 2009-01-12 en_GB dc.description.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. en_GB dc.description.sponsorship Engineering And Physical Sciences Research Council en_GB dc.identifier.uri http://hdl.handle.net/10036/71817 en_GB dc.language.iso en en_GB dc.publisher University of Exeter en_GB dc.subject Hopf-Galois Structure en_GB dc.subject Hopf Algebra en_GB dc.subject Hopf Order en_GB dc.subject Galois Module Structure en_GB dc.subject Algebraic Number Theory en_GB dc.title Hopf-Galois Module Structure Of Some Tamely Ramified Extensions en_GB dc.type Thesis or dissertation en_GB dc.date.available 2009-06-29T10:31:10Z en_GB dc.date.available 2011-01-25T16:55:22Z en_US dc.date.available 2013-03-21T12:04:19Z dc.contributor.advisor Byott, Nigel en_GB dc.publisher.department School Of Engineering, Computer Science And Mathematics en_GB dc.type.degreetitle PhD in Mathematics en_GB dc.type.qualificationlevel Doctoral en_GB dc.type.qualificationname PhD en_GB
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