On Euclidean Ideal Classes.
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 | https://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.type | Thesis | en_US |
dc.description.thesisdegreename | PhD | 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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