Notes on the existence of certain unramified 2-extensions

Abstract

We study the inverse Galois problem with restricted ramification. Let $K$ be an algebraic number field and $G$ be a $2$-group. We consider the question whether there exists an unramified Galois extension $M/K$ with Galois group isomorphic to $G$. We study this question using the theory of embedding problems. Let $L/k$ be a Galois extension and $(\varepsilon): 1\to \mathbf{Z}/2\mathbf{Z}\to E\to \operatorname{Gal} (L/k)\to 1$ a central extension. We first investigate the existence of a Galois extension $M/L/k$ such that the Galois group $\operatorname{Gal} (M/k)$ is isomorphic to $E$ and any finite prime is unramified in $M/L$. As an application, we prove the existence of an unramified extension over cyclic quintic fields with Galois group isomorphic to $32{\Gamma}_5a_2$ under the condition that the class number is even. We also consider the Fontaine-Mazur-Boston Conjecture in the case of abelian $l$-extensions over $\mathbf{Q}$.