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.
Abstract. We improve the known remainder terms for the theorems of Davenport and Heilbronn on the density of discriminants of cubic fields and average 3-torsion in the class group of quadratic fields. This proves analytic continuation of the related Dirichlet series to the left of the line ℜ(s) = 1.
Tate's algorithm for computing K 2 O F for rings of integers in a number field has been adapted for the computer and gives explicit generators for the group and sharp bounds on their order-the latter, together with some structural results on the p-primary part of K 2 O F due to Tate and Keune, gives a proof of its structure for many number fields of small discriminants, confirming earlier conjectural results. For the first time, tame kernels of non-Galois fields are obtained.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.