Nous déterminons sous certaines hypothèses, un système fondamental d’unités du corps non pur $K=\mathbb{Q}\left(\omega \right)$ et de son sous-corps quadratique, où $\omega$ est solution du polynôme$$f\left(X\right)=X^4+d^{-2}{M}_6{X}^2-M_4,$$avec $M_6=D^6+6D^{4}d+9D^2{d}^2+2d^3$, $M_4=D^4+4D^{2}d+2d^2$, $d|D$, $d$, $D\in \mathbb{N}$, non nuls.

Soit $k=\mathbf{Q}\left(\sqrt{-m}\right)$ un corps quadratique imaginaire, soient $k_{\infty }$ et $F$ ses deux ${\mathbf{Z}}_2$-extensions naturelles (la cyclotomique et la prodiédrale), et soit $\check{k}$ son 2-corps de classes de Hilbert. Soient $𝒫$ le complété en 2 de $k$, $\rho =0$ ou 1, égale à 1 si et seulement si tout diviseur impair de $m$ est congru à $\pm 1\mathrm{mod}8$, $\chi =0$ ou 1 le 2-rang de Gal$(k_{\infty }\cap F/k)$, et $t=0,1$ ou 2 le 2-rang de Gal${\check{k}}_{\infty }F\cap \check{k}/k).$ On a $\chi \le \rho$, et des considérations cohomologiques élémentaires nous donnent d’autres contraintes entre $𝒫$, $\chi$ et $t$, mais nous trouvons 2 obstructions supplémentaires de nature...

1. Introduction. Let F be a number field and $O_F$ the ring of its integers. Many results are known about the group $K₂O_F$, the tame kernel of F. In particular, many authors have investigated the 2-Sylow subgroup of $K₂O_F$. As compared with real quadratic fields, the 2-Sylow subgroups of $K₂O_F$ for imaginary quadratic fields F are more difficult to deal with. The objective of this paper is to prove a few theorems on the structure of the 2-Sylow subgroups of $K₂O_F$ for imaginary quadratic fields F. In our Ph.D. thesis (see...

We solve unconditionally the class number one problem for the 2-parameter family of real quadratic fields ℚ(√d) with square-free discriminant d = (an)²+4a for positive odd integers a and n.

General concepts and strategies are developed for identifying the isomorphism type of the second $p$-class group $G=\mathrm{Gal}\left({\mathrm{F}}_p^2\left(K\right)\right|K)$, that is the Galois group of the second Hilbert $p$-class field ${\mathrm{F}}_p^2\left(K\right)$, of a number field $K$, for a prime $p$. The isomorphism type determines the position of $G$ on one of the coclass graphs $𝒢(p,r)$, $r\ge 0$, in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field $K$ and of its $p$-class group ${\mathrm{Cl}}_p\left(K\right)$, the position of $G$ is restricted to certain admissible branches of coclass...