Capitulation des $2$-classes d'idéaux de type $(2,4)$

Abstract

In this paper, we establish two main results which give conditions necessary and sufficient for a $2$-group metabelian such that $G/G^{\prime }$ is of type $(2,4)$ either metacyclic or not. If $G$ is the Galois group of ${\mathbf{k}}_2^{(2)}/\mathbf{k}$ where ${\mathbf{k}}_2^{(2)}$ is the the Second Hilbert $2$-class field of a number field $\mathbf{k}$, we will get results on the problem of capitulation.