Section: New Results

Class groups and other invariants of number fields

Participants : Karim Belabas, Jean-Paul Cerri, Pierre Lezowski.

In [21] , P. Lezowski describes the explicit computation of the Euclidean minimum of a number field. It has been published in Mathematics of Computation.

Ohno and Nakagawa have proved, relations between the counting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakagawa deduced them as consequences of 'extra functional equations' involving the Shintani zeta functions associated to the prehomogeneous vector space of binary cubic forms. In [26] , Henri Cohen, Simon Rubinstein-Salzedo and Frank Thorne generalize their result by proving a similar identity relating certain degree fields with Galois groups $D$ and $F$ respectively, for any odd prime, and in particular we give another proof of the Ohno–Nakagawa relation without appealing to binary cubic forms.

The article [16] by H. Cohen and F. Thorne, H. Cohen on Dirichlet series associated to cubic fields with given resolvent has been published. This article gives an explicit formula for the Dirichlet series ${\sum }_K{\left|\Delta \left(K\right)\right|}^{-s}$, where the sum is over isomorphism classes of all cubic fields whose quadratic resolvent field is isomorphic to a fixed quadratic field $k$.

This work is extended in [15] where H. Cohen give efficient numerical methods for counting exactly the number of $D_{\ell }$ number fields of degree $\ell$ with given quadratic resolvent, for calculating the constants occurring in their asymptotic expansions, and give tables for typical cases.