# On towers of function fields over finite fields

Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2007.

Thesis

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.