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.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/53619 |
| Date | 12 1900 |
| Creators | Harper, John-Paul |
| Contributors | Green, B. W., Stellenbosch University. Faculty of Economic & Management Sciences. Dept. of Business Management. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | Unknown |
| Type | Thesis |
| Format | 106 p. : ill. |
| Rights | Stellenbosch University |
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