Computing Class Groups of Cubic Orders Using Kleinjung’s Sieving Method

Luo, Qinglong

Computing Class Groups of Cubic Orders Using Kleinjung’s Sieving Method

dc.contributor.advisor	Scheidler, Renate
dc.contributor.advisor	Jacobson, Michael J., Jr.
dc.contributor.author	Luo, Qinglong
dc.contributor.committeemember	Bauer, Mark
dc.contributor.committeemember	Tran, Ha
dc.date.accessioned	2024-06-17T14:33:01Z
dc.date.available	2024-06-17T14:33:01Z
dc.date.issued	2024-06-13
dc.description.abstract	We introduce two techniques to generate relations in the computation of class groups of orders of number fields using Buchmann’s index calculus method. In particular, we focus on primitive cubic orders. The first is an adaptation of the homogeneous method from the number field sieve to compute the class groups of a special class of orders known as rationally monogenic orders, which includes primitive cubic orders. The second is a generalisation of recent work by Kleinjung on the quadratic sieve to arbitrarily high degrees. We combine the two into one algorithm, implemented it, and test its performance against MAGMA. The results indicate that our method outperforms MAGMA if the given defining form is skew, and essentially tying MAGMA if the given form is flat.
dc.identifier.citation	Luo, Q. (2024). Computing class groups of cubic orders using Kleinjung’s sieving method (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.
dc.identifier.uri	https://hdl.handle.net/1880/118956
dc.language.iso	en
dc.publisher.faculty	Graduate Studies
dc.publisher.institution	University of Calgary
dc.rights	University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.
dc.subject	class groups
dc.subject	number fields
dc.subject	cubic number fields
dc.subject	Kleinjung
dc.subject	sieving
dc.subject.classification	Education--Mathematics
dc.subject.classification	Computer Science
dc.title	Computing Class Groups of Cubic Orders Using Kleinjung’s Sieving Method
dc.type	doctoral thesis
thesis.degree.discipline	Mathematics & Statistics
thesis.degree.grantor	University of Calgary
thesis.degree.name	Doctor of Philosophy (PhD)
ucalgary.thesis.accesssetbystudent	I do not require a thesis withhold – my thesis will have open access and can be viewed and downloaded publicly as soon as possible.