On towers of function fields over finite fields
Date
2007-03
Authors
Lotter, Ernest Christiaan
Journal Title
Journal ISSN
Volume Title
Publisher
Stellenbosch : University of Stellenbosch
Abstract
Explicit towers of algebraic function fields over finite fields are studied
by considering their ramification behaviour and complete splitting. While
the majority of towers in the literature are recursively defined by a single
defining equation in variable separated form at each step, we consider
towers which may have different defining equations at each step and with
arbitrary defining polynomials.
The ramification and completely splitting loci are analysed by directed
graphs with irreducible polynomials as vertices. Algorithms are exhibited
to construct these graphs in the case of n-step and -finite towers.
These techniques are applied to find new tamely ramified n-step towers
for 1 n 3. Various new tame towers are found, including a family
of towers of cubic extensions for which numerical evidence suggests that
it is asymptotically optimal over the finite field with p2 elements for each
prime p 5. Families of wildly ramified Artin-Schreier towers over small
finite fields which are candidates to be asymptotically good are also considered
using our method.
Description
Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2007.
Keywords
Class field towers, Algebraic fields, Function algebras, Dissertations -- Mathematics, Theses -- Mathematics