Infrastructure, Arithmetic, and Class Number Computations in Purely Cubic Function Fields of Characteristic at Least 5

One of the more difficult and central problems in computational algebraic number theory is the computation of certain invariants of a field and its maximal order. In this thesis, we consider this problem where the field in question is a purely cubic function field, K/Fq(x), with char(K) ≥ 5. In addition, we will give a divisor-theoretic treatment of the infrastructures ofK, including a description of its arithmetic, and develop arithmetic on the ideals of the maximal order, O, of K. Historically, the infrastructure, RC, of an ideal class, C ∈ Cl(O) has been defined as a set of reduced ideals in C. However, we extend work of Paulus and Rück [PR99] and Jacobson, Scheidler, and Stein [JSS07b] to define RC as a certain subset of the divisor class group, JK , of a cubic function field, K, specifically, the subset of distinguished divisors whose classes map to C via JK → Cl(O). Our definition of distinguished generalizes the same notion by Bauer for purely cubic function fields of unit rank 0 [Bau04] to those of unit rank 1 and 2 as well. Further, we prove a bijection between RC, as a set of distinguished divisors, and the infrastructure of C defined by “reduced” ideals, as in [Sch00, SS00, Sch01, LSY03, Sch04]. We describe the arithmetic on RC, providing new results on the baby step and giant step operations and generalizing notions of the inverse of a divisor in R[O] from quadratic infrastructures in [JSS07b] to cubic infrastructures. We also give algorithms to compute the various operations. For the infrastructure arithmetic, as well as for computing in Cl(O), we derive ideal arithmetic for any purely cubic function field, K, with char(K) 6= 2, generalizing work of Scheidler [Sch01] and Bauer [Bau04]. In addition, we show how to determine the unique distinguished ideal in a given ideal class in the case that K has unit rank 0, extending results of Bauer [Bau04] from cubic function fields defined by a non-singular curve to those defined by a singular curve as well. For the ideal arithmetic and reduction methods, we provide algorithms as well. Finally, we describe methods to compute the divisor class number, h, of K, and in the case that O has unit rank 1 or 2, the regulator and ideal class number of O as well. A method of Scheidler and Stein [SS07, SS08] determines sharper upper and lower bounds on h, for a given cubic function field, than those given by the Hasse-Weil Theorem. We then employ Shanks’ Baby Step-Giant Step algorithm [Sha71] and Pollard’s Kangaroo method [Pol78], to search this interval and compute the desired invariants for purely cubic function fields of unit rank 0 and 1. The total complexity of the method to compute these invariants is O ( q(2g−1)/5+ε(g) ) ideal operations as q → ∞, where 0 ≤ ε(g) ≤ 1/5. With this approach, we computed the 28 decimal digit divisor class numbers of two purely cubic function fields of genus 3: one of unit rank 0 and one of unit rank 1. We also computed the 25 decimal digit divisor class numbers of two purely cubic function fields of genus 4: one of unit rank 0 and one of unit rank 1. In the unit rank 1 examples, we factored the divisor class numbers