A fast algorithm to compute cubic fields

Abstract

Abstract. We present a very fast algorithm to build up tables of cubic fields. Real cubic fields with discriminant up to 10 11 and complex cubic fields down to −10 11 have been computed.The classification of quadratic fields up to isomorphism is trivial: they are uniquely characterized by their discriminant, and we can compute tables as soon as we know how to test if an integer is squarefree and how to check some simple congruence modulo 16. We intend to show that cubic fields are essentially as easy to deal with, and we will get a canonical representation for them. Contrary to the quadratic case, the treatment depends on the signature but, the fundamental ideas being the same, we shall expose as much as we can before splitting cases.Almost all results in this paper are either ancient or elementary. I would like to thank Professor H. Cohen for his interest when I first mentioned what I thought was a trivial application of some well known results. Moreover, his careful reading of successive drafts of this work and the many questions he had about it were most helpful in giving it its present shape.

Computing cubic fields in quasi-linear time

“…When no explicit reference has been given, we refer the curious reader not wishing to consider the proofs as (easy) exercises to [1].…”

Tabulation of Cubic Function Fields with Imaginary and Unusual Hessian

“…Hence, the membership test for U is more efficient than its counterpart for integral forms. The algorithms presented here have some of the same advantages as Belabas' algorithm [2]. In particular, there is no need to check for irreducibility of binary cubic forms lying in U, no need to factor the discriminant, and no need to keep all fields found so far in memory.…”

“…Analogous to [2], one can deduce that any two equivalent reduced imaginary forms are equal, so equivalence classes of such forms can be efficiently identified by their unique reduced representative. …”

“…In 1997, Belabas [2] presented an algorithm for tabulating all non-isomorphic cubic number fields of discriminant D with |D| ≤ X for any X > 0. The results make use of the reduction theory for binary cubic forms with integral coefficients.…”

“…That is, we present a method for tabulating all cubic function fields over a fixed finite field up to a given upper bound on the degree of the discriminant, using the theory for binary cubic forms with coefficients in F q [t], where F q is a finite field with char(F q ) = 2, 3. While some of the ideas of [2] translate essentially directly from number fields to function fields, there are in fact a number of obstructions to a straightforward adaptation of Belabas' algorithm [2] to the function field setting. Firstly, there is a very simple connection between the signatures of cubic and quadratic number fields of the same discriminant D, which are simply characterized as real or complex/imaginary according to whether D > 0 or D < 0.…”