On Euclidean Ideal Classes.

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dc.contributor.author Graves, Hester K. en_US
dc.date.accessioned 2009-09-03T14:53:54Z
dc.date.available NO_RESTRICTION en_US
dc.date.available 2009-09-03T14:53:54Z
dc.date.issued 2009 en_US
dc.date.submitted 2009 en_US
dc.identifier.uri http://hdl.handle.net/2027.42/63828
dc.description.abstract In 1979, H.K.Lenstra generalized the idea of Euclidean algorithms to Euclidean ideal classes. If a domain has a Euclidean algorithm, then it is a principal ideal domain and has a trivial class group; if a Dedekind domain has a Euclidean ideal class, then it has a cyclic class group gen- erated by the Euclidean ideal class. Lenstra showed that if one assumes the generalized Riemann hypothesis and a number field has a ring of in- tegers with infinitely many units, then said ring has cyclic class group if and only if it has a Euclidean ideal class. Malcolm Harper’s dissertation built up general machinery that allows one to show a given ring of integers (with infinitely many units) of a number field with trivial class group is a Euclidean ring. In order to build the machinery, Harper used the Large Sieve and the Gupta-Murty bound. This dissertation generalizes Harper’s work to the Euclidean ideal class setting. In it, there is general machinery that allows one to show that a number field with cyclic class group and a ring of integers with infinitely many units has a Euclidean ideal class. In order to build this machinery, the Large Sieve and the Gupta-Murty bound needed to be generalized to the ideal class situation. The first required class field theory; the second required several asymptotic results on the sizes of sets of k-tuples. en_US
dc.format.extent 615496 bytes
dc.format.extent 1373 bytes
dc.format.mimetype application/pdf
dc.format.mimetype text/plain
dc.language.iso en_US en_US
dc.subject Euclidean Ideal Class en_US
dc.subject Large Sieve en_US
dc.subject Gupta-Murty Bound en_US
dc.subject Euclidean en_US
dc.subject Class Group en_US
dc.subject Cyclic en_US
dc.title On Euclidean Ideal Classes. en_US
dc.description.thesisdegreename Ph.D. en_US
dc.description.thesisdegreediscipline Mathematics en_US
dc.description.thesisdegreegrantor University of Michigan, Horace H. Rackham School of Graduate Studies en_US
dc.contributor.committeemember Ramsey, Nicholas Adam en_US
dc.contributor.committeemember Hall, Christopher J. en_US
dc.contributor.committeemember Jonsson, Mattias en_US
dc.contributor.committeemember Lagarias, Jeffrey C. en_US
dc.contributor.committeemember Levitsky, Melvyn en_US
dc.contributor.committeemember Skinner, Christopher M. en_US
dc.subject.hlbsecondlevel Mathematics en_US
dc.subject.hlbtoplevel Science en_US
dc.description.bitstreamurl http://deepblue.lib.umich.edu/bitstream/2027.42/63828/1/gravesh_1.pdf
dc.owningcollname Dissertations and Theses (Ph.D. and Master's)
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