Algorithmic Aspects of Cubic Function Fields

This paper presents an investigative account of arbitrary cubic function fields. We present an elementary classification of the signature of a cubic extension of a rational function field of finite characteristic at least five; the signature can be determined solely from the coefficients of the defining curve. We go on to study such extensions from an algorithmic perspective, presenting efficient arithmetic of reduced ideals in the maximal order as well as algorithms for computing the fundamental unit(s) and the regulator of the extension.