On Galois groups of \(p\)-closed algebraic number fields with restricted ramification. (English) Zbl 0715.11065

Wingberg, Kay

J. Reine Angew. Math. 400, 185-202 (1989).

Assume the algebraic number field \(k\) to contain the \(2p\)-th root of unity. Then, roughly speaking, the following is shown. Let \(k_S(p)\) be the maximal \(p\)-extension of \(k\) which is unramified outside a finite set \(S\) of primes containing all the ones above \(p\) and \(\infty\), and let \(G_S\) be the corresponding Galois group \(\mathrm{Gal}(k_S(p)/k)\). Then \(G_S\) is either a duality group, i.e. a group whose cohomology satisfies certain duality relations, or it is a free pro-\(p\) product of decomposition groups and a free pro-\(p\) group.
The proof rests heavily on L. V. Kuz’min’s work [Math. USSR, Izv. 9(1975), 693–726 (1976); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 39, 739–772 (1975; Zbl 0334.12027)]; in an appendix the author gives a nice account of this.

Reviewer: Jürgen Ritter (Augsburg)

MSC:

11R32	Galois theory
11R34	Galois cohomology

Keywords:

maximal p-extension; Galois group; duality group; pro-p group

Citations:

Zbl 0334.12027