Hopf-Galois Module Structure Of Some Tamely Ramified Extensions

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Hopf-Galois Module Structure Of Some Tamely Ramified Extensions

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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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