Descent of Kummerian Galois \(p\)-extensions. (Descente de \(p\)-extensions galoisiennes kummériennes.) (French) Zbl 0876.12004

Let \(K/J\) be any fixed algebraic extension of fields. The author poses the following problem that she calls “the Galois descent problem”: for a given Galois extension \(E/K\), does there exist a field \(D\subseteq E\) with \(D/J\) a Galois extension such that \(D\cap K=J\) and the composition \(DK=E\)? A first Galois descent result, stated in other terms, was proved by G. Brattström [Math. Scand. 65, 165-174 (1989; Zbl 0707.12001)] for \(p\)-extensions of degree \(p^3\) (\(p\) an odd prime). In the paper under review, the author generalizes this result by succeeding the “\(p\)-cyclotomic Galois descent” of non-abelian \(p\)-extensions, over a base field \(K\) containing the group \(\mu_p\) of \(p\)-th roots of unity, explicitly constructed by R. Massy [J. Algebra 109, 508-535 (1987; Zbl 0625.12011)]. She provides the Galois descent from these \(p\)-extensions, with fixed cohomology class, over any subfield \(J\subseteq K\) such as \(J\cap \mu_p=\{1\}\).
Reviewer’s remark: in the theorem 4.3, it is possible to delete the power \(d^{-1}\) in the primitive element formulas of the fields \(N'\) and \(D\). This power is just necessary to get the right 2-cocycle.


12F10 Separable extensions, Galois theory
11R32 Galois theory
11S20 Galois theory
Full Text: DOI EuDML