The class number one problem in function fields

[摘要] ENGLISH ABSTRACT:In this dissertation I investigate the class number one problem in function fields. Moreprecisely I give a survey of the current state of research into extensions of a rational functionfield over a finite field with principal ring of integers. I focus particularly on the quadraticcase 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 ofquadratic extensions of the rational numbers and the corresponding rings of integers. Morerecently however work has been done in the situation of function fields in which the arithmeticis very similar.I begin with an introduction into the arithmetic in function fields over a finite field andprove the analogies of many of the classical results. I then proceed to demonstrate how thealgebra and arithmetic in function fields can be interpreted geometrically in terms of curvesand introduce the associated geometric language. After presenting some conjectures, I proceedto give a survey of known results in the situation of quadratic function fields. I present alsoa few results of my own in this section. Lastly I state some recent results regarding arbitraryextensions of a rational function field with principal ring of integers and give some heuristicresults regarding class groups in function fields.

[发布机构] Stellenbosch University