# Séminaires : Séminaire Théorie des Nombres

 Equipe(s) : tn, Responsables : Ziyang Gao, Marc Hindry, Bruno Kahn, João Pedro P. dos Santos Email des responsables : ziyang.gao@imj-prg.fr Salle : Adresse : Description http://www.imj-prg.fr/tn/STN/stnj.html

 Orateur(s) Henri Cohen - , Titre Quartic and $D_\ell$ fields of degree $\ell$ with givenresolvent. Date 08/04/2013 Horaire 14:00 à 16:00 Diffusion Résume If $L$ is a degree $\ell$field with Galois group of Galois closure $D_\ell$, then $L$ has aquadratic resolvent $k$, and $\disc(L)=(\disc(k)f(L))^{(\ell-1)/2}$ fora suitable integer $f(L)$. We give a completely explicit formula forthe Dirichlet series $\sum_L f(L)^{-s}$ in terms of Euler productsattached to a finite number of auxiliary fields. This has applicationsboth in the exact counting and in the asymptotics of such degree $\ell$fields. The same is also done for quartic fields with Galois closure$A_4$ or $S_4$ and given cubic resolvent. Salle Adresse