A fast algorithm to compute cubic fields

We present a very fast algorithm to build up tables of cubic fields. Real cubic fields with discriminant up to 1011 and complex cubic fields down to −1011 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.