We show that for 100% of the odd, squarefree integers n > 0 the 4-rank of Cl(Q(i, √ n)) is equal to ω 3 (n) − 1, where ω 3 is the number of prime divisors of n that are 3 modulo 4.

The software package KANT V4 for computations in algebraic number fields is now available in version 4. In addition a new user interface has been released. We will outline the features of this new software package.

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This paper announces the creation of a database for number fields. It describes the contents and the methods of access, indicates the origin of the polynomials, and formulates the aims of this collection of fields.

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In this Note we give a counter example to a conjecture of Malle which predicts the asymptotic behavior of the counting functions for field extensions with given Galois group and bounded discriminant. To cite this article: J. Klüners, C. R. Acad. Sci. Paris, Ser. I 340 (2005).  2005 Académie des sciences. Published by Elsevier SAS. All rights reserved. Résumé Un contre-exemple à la conjecture de Malle sur le nombre de corps de discriminant borné. Dans cette Note, nous donnons un contre-exemple à une conjecture de Malle, qui prédit le comportement asymptotique du nombre de corps de clôture galoisienne fixée et discriminant borné. Pour citer cet article : J. Klüners, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Let K be a global field and f ∈ K[X] be a polynomial. We present an efficient algorithm which factors f in polynomial time.

Computational Galois theory, in particular the problem of computing the Galois group of a given polynomial, is a very old problem. Currently, the best algorithmic solution is Stauduhar's method. Computationally, one of the key challenges in the application of Stauduhar's method is to find, for a given pair of groups H < G, a G-relative H-invariant, that is a multivariate polynomial F that is H-invariant, but not G-invariant. While generic, theoretical methods are known to find such F , in general they yield impractical answers. We give a general method for computing invariants of large degree which improves on previous known methods, as well as various special invariants that are derived from the structure of the groups. We then apply our new invariants to the task of computing the Galois groups of polynomials over the rational numbers, resulting in the first practical degree independent algorithm.