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Soit $k$ un corps de nombres et soient $T$ et $S$ deux ensembles finis de places de $k$ ; on peut définir la tour de Hilbert de $k$, $T$-ramifiée modérée, $S$-décomposée. Ceci permet d’obtenir, par exemple, la notion de tour de Hilbert au sens classique et de tour de Hilbert au sens restreint. On donne alors d’une part, un critère de finitude de cette nouvelle tour, critère construit à partir d’un résultat d’Odlyzko, puis d’autre part deux critères de non-finitude, le premier étant une conséquence d’un résultat...

In this paper, we study the cohomological dimension of groups $G_S:=Gal({𝕂}_S/𝕂)$, where ${𝕂}_S$ is the maximal pro-$p$-extension of a number field $𝕂$, unramified outside a finite set $S$ of places of $𝕂$. This dimension is well-understood only when $S$ contains all places above $p$; in the case where only some of the places above $p$ are contained in $S$, one can still obtain some results if ${𝕂}_S/𝕂$ contains at least one ${\mathbb{Z}}_p$-extension ${𝕂}_{\infty }/𝕂$. Indeed, in that case, the study of the ${\mathbb{Z}}_p\left[[Gal({𝕂}_{\infty }/𝕂)]\right]$-module $Gal{({𝕂}_S/{𝕂}_{\infty })}^{ab}$ allows one to give sufficient conditions for the pro-$p$-group...

Much recent progress in the 2-class field tower problem revolves around demonstrating infinite such towers for fields – in particular, quadratic fields – whose class groups have large 4-ranks. Generalizing to all primes, we use Golod-Safarevic-type inequalities to analyse the source of the $p^2$-rank of the class group as a quantity of relevance in the $p$-class field tower problem. We also make significant partial progress toward demonstrating that all real quadratic number fields whose class groups...