Arithmetic Problems in Cubic and Quartic Function Fields

TL;DR

This thesis develops a new theory for analyzing the signatures of places in cubic and quartic function fields, avoiding p-adic methods and Cardano's formulae, and introduces constructions of specific function fields with applications to divisor class numbers.

Contribution

It presents a novel, p-adic-free approach to signatures in cubic and quartic function fields and constructs fields with explicit fundamental systems and new divisor class number methods.

Findings

New theory for signatures in cubic and quartic function fields

Construction of cubic function fields with known fundamental systems

A new approach for determining divisor class numbers

Abstract

One of the main themes in this thesis is the description of the signature of both the infinite place and the finite places in cubic function fields of any characteristic and quartic function fields of characteristic at least 5. For these purposes, we provide a new theory which can be applied to cubic and quartic function fields and to even higher dimensional function fields. One of the striking advantages of this theory to other existing methods is that is does not use the concept of p-adic completions and we can dispense of Cardano's formulae. Another key result comprises the construction of cubic function fields of unit rank 1 and 2, with an obvious fundamental system. One of the main ingredients for such constructions is the definition of the maximum value. This definition is new and very prolific in the context of finding fundamental systems. We conclude the thesis with miscellaneous results on the divisor class number h, including a new approach for finding divisors of h.