On the pro-\(p\)-extensions with restricted ramification over the cyclotomic \(\mathbb Z_p\)-extension of a number field
This article, first part of the author's thesis [\textit{L. Salle}, Présentations de groupes de Galois de pro-\(p\)-extensions de corps de nombres, Thèse de l'Université Toulouse III (2008)], deals with the maximal pro-\(p\)-extensions with restricted ramification over the cyclotomic \({\mathbb Z}_p\)-extension over a number field. The author investigates their Galois group, focusing first on the \({\mathbb Z}_p\)-rank of their abelianization and then on their minimal numbers of generators and relations. The author uses class field theory and Koch Galois theory of pro-\(p\)-extensions in [\textit{H. Koch}, Galois theory of \(p\)-extensions. Berlin: Springer (2002; Zbl 1023.11002)]. Let \(k\) a field of algebraic numbers, \(p\) a prime, \(k_\infty\) the cyclotomic \({\mathbb Z}_p\)-extension of \(k\), and \(S\) a finite set of places of \(k\). In this article the author examines the maximal \(S\)-ramified pro-\(p\)-extension \(\overline{L}_S(k_\infty)\) of \(k_\infty\) and the maximal abelian sub-extension \(\overline{L}_S(k_\infty)^{ab}\) of \(\overline{L}_S(k_\infty)/k\). When \(S=\emptyset\), the corresponding maximal abelian extension, denoted \(\overline{L}(k_\infty)^{ab}\) is the cyclotomically ramified extension, following the article [\textit{J.-F. Jaulent} and \textit{J. Sands}, Compos. Math. 99, No. 3, 325--341 (1995; Zbl 0869.11084)]. In the other hand, when \(S\) contains all the places over \(p\), this extension is precisely the maximal \(S\)-ramified pro-\(p\)-extension. In a first part, the author considers the \({\mathbb Z}_p\)-free part of the group \(\text{Gal}(\overline{L}_S(k_\infty)^{ab}/k)\) caracterized as a \({\mathbb Z}_p\)-module by its \({\mathbb Z}_p\)-rank. Several results exists on this \({\mathbb Z}_p\)-rank in Jaulent and Sands article [loc. cit.] when \(S=\emptyset\). This article systematizes this approach when \(S\not=\emptyset\) with the use of the \(p\)-adic theory as developed by \textit{J.-F. Jaulent} in [J. Théor. Nombres Bordx. 10, No. 2, 355--397 (1998; Zbl 0938.11052)]. In a second part of the article, following Koch [loc. cit.] arguments for the study of maximal pro-\(p\)-extensions with restricted ramifications, the author examines the group \(\text{Gal}(\overline{L}_S(k_\infty)/k)\) through its generators number (which is smaller or equal to the previous \({\mathbb Z}_p\)-rank) and its number of relations. The tools of this second part are class field theory and Galois cohomology.
scientific article; zbMATH DE number 5543173