A note on free pro-\(p\)-extensions of algebraic number fields (Q1311054)

The author studies the maximal rank \(\rho\) of free pro-\(p\) Galois groups \(F_ \rho\) over a fixed algebraic number field \(k\) for a fixed prime \(p\). Such an \(F_ \rho\) is necessarily a quotient of \(G_{S_ p}\), the Galois group over \(k\) of the maximal pro-\(p\)-extension of \(k\) unramified outside \(p\). By class field theory, the rank of \(G_{S_ p}^{ab}\) is \(1+r_ 2+\delta\), where \(r_ 2\) denotes the number of complex places of \(k\), and \(\delta\) the defect of Leopoldt's conjecture. The main results are the following: 1) \(\rho\leq 1+r_ 2\) if the ``weak Leopoldt conjecture'' holds for all \(\mathbb{Z}_ p\)-extensions of \(k\). 2) There exist \(k\), \(p\) such that \(\rho<1+r_ 2\). Such examples come from number fields \(k\) for which \(G_{S_ p}\), under rather strong conditions, is a quotient of a free pro-\(p\)-product of decomposition groups and a free group [see \textit{K. Wingberg}, J. Reine Angew. Math. 400, 185-202 (1989; Zbl 0715.11065)].