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

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.


11R32 Galois theory
11R34 Galois cohomology


Zbl 0334.12027
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