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_GB |
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_GB |
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 |