## 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.

### MSC:

 12F10 Separable extensions, Galois theory 11R32 Galois theory 11S20 Galois theory

### Citations:

Zbl 0707.12001; Zbl 0625.12011
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