On Euclidean Ideal Classes.

Show simple item record

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)
 Show simple item record

This item appears in the following Collection(s)


Search Deep Blue

Advanced Search

Browse by

My Account

Information

Coming Soon


MLibrary logo