Algorithmic aspects of cubic function fields, in Algorithmic number theory

Abstract. This paper presents an investigative account of arbitrary cu-bic function fields. We present an elementary classification of the signa-ture 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 algo-rithmic 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.