Generic Galois extensions and problems in field theory. (English) Zbl 0484.12004

Saltman, David J.

Adv. Math. 43, 250-283 (1982).

In Galois theory, the study of Kummer extensions and Artin-Schreier polynomials have a common aspect. Namely, that all Galois extensions with a fixed group are described as specializations of a single ”general” extension with ”independent” indeterminates. In this paper, this idea is taken up and formalized in the notion of a generic Galois extension.
In one direction, this paper shows that several classes of groups have generic Galois extensions. In another direction, the existence of generic Galois extensions is shown to be related to Noether’s problem on invariants, the Grunwald-Wang theorem of number theory, and lifting questions for Galois extensions over local rings. As a consequence, one has a natural, elementary proof of a chunk of the Grunwald-Wang theorem (extending an earlier elementary proof) which also applies to more general valued fields and some non-ebelian groups. One also has an elementary proof of a known counterexample to Noether’s problem. Finally, the existence of a generic Galois extension is shown to be equivalent to a lifting property for Galois extensions over local algebras. In later papers, by this author, some of these ideas are further refined and extended.

Reviewer: David J. Saltman

MSC:

12F10	Separable extensions, Galois theory
20B25	Finite automorphism groups of algebraic, geometric, or combinatorial structures
11R18	Cyclotomic extensions
12G05	Galois cohomology

Keywords:

generic Galois extension; Grunwald-Wang theorem; lifting; Noether’s problem

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