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The class number one problem in function fields
Thesis (MComm)--Stellenbosch University, 2003.
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2012
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| _version_ | 1867613941587247104 |
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| access_status_str | Open Access |
| author | Harper, John-Paul |
| author2 | Green, B. W. |
| author_browse | Green, B. W. Harper, John-Paul |
| author_facet | Green, B. W. Harper, John-Paul |
| author_sort | Harper, John-Paul |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MComm)--Stellenbosch University, 2003. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/53619 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:44:08.546Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2012 |
| publishDateRange | 2012 |
| publishDateSort | 2012 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/53619 The class number one problem in function fields Harper, John-Paul Green, B. W. Stellenbosch University. Faculty of Economic & Management Sciences. Dept. of Business Management. Geometric function theory Functional analysis Algebraic functions Dissertations -- Mathematics Theses -- Mathematics Thesis (MComm)--Stellenbosch University, 2003. ENGLISH ABSTRACT: In this dissertation I investigate the class number one problem in function fields. More precisely I give a survey of the current state of research into extensions of a rational function field over a finite field with principal ring of integers. I focus particularly on the quadratic case and throughout draw analogies and motivations from the classical number field situation. It was the "Prince of Mathematicians" C.F. Gauss who first undertook an in depth study of quadratic extensions of the rational numbers and the corresponding rings of integers. More recently however work has been done in the situation of function fields in which the arithmetic is very similar. I begin with an introduction into the arithmetic in function fields over a finite field and prove the analogies of many of the classical results. I then proceed to demonstrate how the algebra and arithmetic in function fields can be interpreted geometrically in terms of curves and introduce the associated geometric language. After presenting some conjectures, I proceed to give a survey of known results in the situation of quadratic function fields. I present also a few results of my own in this section. Lastly I state some recent results regarding arbitrary extensions of a rational function field with principal ring of integers and give some heuristic results regarding class groups in function fields. AFRIKAANSE OPSOMMING: In hierdie tesis ondersoek ek die klasgetal een probleem in funksieliggame. Meer spesifiek ondersoek ek die huidige staat van navorsing aangaande uitbreidings van 'n rasionale funksieliggaam oor 'n eindige liggaam sodat die ring van heelgetalle 'n hoofidealgebied is. Ek kyk in besonder na die kwadratiese geval, en deurgaans verwys ek na die analoog in die klassieke getalleliggaam situasie. Dit was die beroemde wiskundige C.F. Gauss wat eerste kwadratiese uitbreidings van die rasionale getalle en die ooreenstemende ring van heelgetalle in diepte ondersoek het. Onlangs het wiskundiges hierdie probleme ook ondersoek in die situasie van funksieliggame oor 'n eindige liggaam waar die algebraïese struktuur baie soortgelyk is. Ek begin met 'n inleiding tot die rekenkunde in funksieliggame oor 'n eindige liggaam en bewys die analogie van baie van die klassieke resultate. Dan verduidelik ek hoe die algebra in funksieliggame geometries beskou kan word in terme van kurwes en gee 'n kort inleiding tot die geometriese taal. Nadat ek 'n paar vermoedes bespreek, gee ek 'n oorsig van wat alreeds vir quadratiese funksieliggame bewys is. In hierdie afdeling word 'n paar resultate van my eie ook bewys. Dan vermeld ek 'n paar resultate aangaande algemene uitbreidings van 'n rasionale funksieliggaam oor 'n eindige liggaam waar die van ring heelgetalle 'n hoofidealgebied is. Laastens verwys ek na 'n paar heurisitiese resultate aangaande klasgroepe in funksieliggame. Masters 2012-08-27T11:35:33Z 2012-08-27T11:35:33Z 2003-12 Thesis http://hdl.handle.net/10019.1/53619 en_ZA Stellenbosch University 106 p. : ill. application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Geometric function theory Functional analysis Algebraic functions Dissertations -- Mathematics Theses -- Mathematics Harper, John-Paul The class number one problem in function fields |
| title | The class number one problem in function fields |
| title_full | The class number one problem in function fields |
| title_fullStr | The class number one problem in function fields |
| title_full_unstemmed | The class number one problem in function fields |
| title_short | The class number one problem in function fields |
| title_sort | class number one problem in function fields |
| topic | Geometric function theory Functional analysis Algebraic functions Dissertations -- Mathematics Theses -- Mathematics |
| url | http://hdl.handle.net/10019.1/53619 |
| work_keys_str_mv | AT harperjohnpaul theclassnumberoneprobleminfunctionfields AT harperjohnpaul classnumberoneprobleminfunctionfields |