La composition de Gauss donne une structure de groupe aux orbites de formes quadratiques binaires entières de discriminant $D$, sous l’action de ${\mathrm{SL}}_2$ par changement de variable, essentiellement le groupe des classes de l’ordre quadratique de discriminant $D$. Les domaines fondamentaux associés permettent calculs explicites et évaluation d’ordres moyens. Je présenterai les lois de composition supérieures découvertes par M. Bhargava à partir de la classification des espaces vectoriels préhomogènes réguliers,...

Let $p$ be an odd prime, $r$ be a primitive root modulo $p$ and $r_i\equiv r^i(modp)$ with $1\le r_i\le p-1$. In 2007, R. Queme raised the question whether the $\ell$-rank ($\ell$ an odd prime $\ne p$) of the ideal class group of the $p$-th cyclotomic field is equal to the degree of the greatest common divisor over the finite field ${𝔽}_{\ell }$ of $x^{(p-1)/2}+1$ and Kummer’s polynomial $f\left(x\right)={\sum }_{i=0}^{p-2}{r}_{-i}{x}^i$. In this paper, we shall give the complete answer for this question enumerating a counter-example.

For an algebraic number field $K$ with $3$-class group ${\mathrm{Cl}}_3\left(K\right)$ of type $(3,3)$, the structure of the $3$-class groups ${\mathrm{Cl}}_3\left(N_i\right)$ of the four unramified cyclic cubic extension fields $N_i$, $1\le i\le 4$, of $K$ is calculated with the aid of presentations for the metabelian Galois group ${\mathrm{G}}_3^2\left(K\right)=\mathrm{Gal}\left({\mathrm{F}}_3^2\left(K\right)\right|K)$ of the second Hilbert $3$-class field ${\mathrm{F}}_3^2\left(K\right)$ of $K$. In the case of a quadratic base field $K=\mathbb{Q}\left(\sqrt{D}\right)$ it is shown that the structure of the $3$-class groups of the four $S_3$-fields $N_1,...,N_4$ frequently determines the type of principalization of the $3$-class group of $K$ in $N_1,...,N_4$. This provides...