Finiteness of \(T\)-tamely ramified and \(S\)-decomposed towers and \(p\)-towers
If \(T\) and \(S\) are (finite, disjoint) sets of places of a number field \(k\), let \(k_1= k_1^{S,T}\) be the greatest abelian extension of \(k\), unramified outside \(T\), moderately ramified on \(T\), where places of \(S\) split completely. Denoting by \(S_1\) and \(T_1\) the places of \(k_1\) lying over \(S\) and \(T\), one can iterate this construction, obtaining the \(T-S\)-hilbert tower of \(k\). The author begins with a finiteness criterion of the \(p\)-part of this tower (for fixed prime number \(p)\). Namely, if the \(p\)-Sylow such group of the Galois group \(\text{Gal} (k_1/k) = cl^{S,T} (k)\) is cyclic, then the tower is finite (and in fact stops at the first stage). Then, the author gives two nonfiniteness criteria of the tower (as usual, both involving the Golod-Shafarevich theorem on finite \(p\)-groups), the second one involving Jaulent's genus theory. Many examples are given. For instance, the field \(\mathbb{Q}(\sqrt{53.131})\) has infinite tower in the narrow sense (i.e. for \(T\) and \(S\) empty), but finite tower in the classical sense (i.e., \(T\) and \(S\) empty), \(S=S^\infty)\). The author concludes with considerations (using Odlyzko bounds) on the discriminant of fields having infinite tower.
scientific article; zbMATH DE number 912498