Classes de Steinitz d’extensions galoisiennes non ramifiées
Streszczenie
Let $K$ be a number field and ${\rm Cl}_{K}$ its class group. Let $\Gamma $ be a finite group. We denote by $ R_{\rm nr}(K,\Gamma )$ the subset of ${\rm Cl}_{K}$ formed by the classes which are realizable as Steinitz classes of Galois extensions of $K$, unramified at the finite places of $K$ and having a Galois group isomorphic to $\Gamma $. If $\Gamma $ is abelian and the narrow class number of $K$ is prime to the order of $\Gamma $, then $R_{\rm nr}(K,\Gamma )=\emptyset $. In the present article, for any $\Gamma $ we consider the set $R’_{\rm nr}(K,\Gamma ):=\lbrace 1 \rbrace \cup R_{\rm nr}(K,\Gamma )$. We prove that $R’_{\rm nr}(K, \mathbb {Z}/2\mathbb {Z})$ is a subgroup of ${\rm Cl}_{K,2}:=\lbrace c\in {\rm Cl}_{K}\mid c^{2}=1\rbrace $; furthermore, it is equal to ${\rm Cl}_{K,2}$ under a certain assumption on $K$. Using this result we show that $R’_{\rm nr}(K,\Gamma )$ is a subgroup of $R’_{\rm nr}(K, \mathbb {Z}/2\mathbb {Z})$ if $\Gamma $ either has odd order, or has a noncyclic $2$-Sylow subgroup (for instance $\Gamma $ is a nonabelian $2$-group, $\Gamma =S_n$ or $A_n$ with $n\geq 4$), or has a normal cyclic $2$-Sylow subgroup (for instance $\Gamma $ is nilpotent of even order).
