| 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. |
