A note on free pro-\(p\)-extensions of algebraic number fields. (English) Zbl 0784.11052

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 K. Wingberg, J. Reine Angew. Math. 400, 185-202 (1989; Zbl 0715.11065)].


11R32 Galois theory
11R37 Class field theory
11R34 Galois cohomology


Zbl 0715.11065
Full Text: DOI Numdam EuDML


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