Computing Cubic Fields in Quasi-Linear Time

Cubic fields (over the rationals) are the simplest non-Galois number fields and thus should be the ideal testing ground for most general “density” conjectures, such as the Cohen-Martinet heuristics. We present an efficient algorithm to generate them, up to a given discriminant bound, which we hope will prove a useful tool in their computational exploration. It all originates from the seminal paper [4] by Davenport and Heilbronn and some reduction theory as was already known to Hermite. When no explicit reference has been given, we refer the curious reader not wishing to consider the proofs as (easy) exercises to [1]. The rationale is as follows: to a given cubic field, we associate first a class of binary cubic forms, which shares the same discriminant, and then a canonical representative in the class. The essential point is that we have an explicit description of the image of this mapping, the set of companion forms, which behaves nicely from the algorithmic point of view.