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...

The so-called Lifted Root Number Conjecture is a strengthening of Chinburg’s $\Omega \left(3\right)$- conjecture for Galois extensions $K/F$ of number fields. It is certainly more difficult than the $\Omega \left(3\right)$-localization. Following the lead of Ritter and Weiss, we prove the Lifted Root Number Conjecture for the case that $F=\mathbb{Q}$ and the degree of $K/F$ is an odd prime, with another small restriction on ramification. The very explicit calculations with cyclotomic units use trees and some classical combinatorics for bookkeeping. An important...

Soit $K$ un corps de nombres contenant ${\mu }_p$ et muni d’un groupe d’automorphismes $G$ d’ordre étranger à $p$ ; pour toute représentation ${𝔽}_p$-irréductible $V_{\chi }$ de $G$, de caractère $\chi$, et tout $G$-module $M$, soit rg${}_{\chi }\left(M\right)$ l’entier $r$ maximum tel que $M/M^p$ contienne $V_{\chi }^r$. Nous établissons par exemple la formule générale explicite suivante :$$r{g}_{{\chi }^{*}}\left(C{\ell }_T^S\right)-r{g}_{\chi }\left(C{\ell }_S^T\right)={\rho }_{\chi }(T,S),$$où $T$ et $S$ sont des ensembles finis disjoints de places de $K$ tels que $T\cup S$ contienne les places au-dessus de $p\infty$, où $C{\ell }_T^S$ est le groupe de classes généralisées qui correspond, par le corps de classes, au...

Nous établissons les résultats fondamentaux de la théorie $\ell$-adique globale du corps de classes pour les corps de nombres.

We give exhaustive list of biquadratic fields $K=\mathbb{Q}(i,\sqrt{m})$ and $K=\mathbb{Q}(\sqrt{2},\sqrt{m})$ without $2$-exotic symbol, i.e. for which the $2$-rank of the Hilbert kernel (or wild kernel) is zero. Such $K=\mathbb{Q}(i,\sqrt{m})$ are logarithmic principals [J3]. We detail an exemple of this technical numerical exploration and quote the family of theories and results we utilize. The $2$-rank of tame, regular and wild kernel of $K$-theory are connected with local and global problem of embedding in a $Z_2$-extension. Global class field theory can describe the $2$-rank of the Hilbert...