Generic Galois extensions.
We define the notion of a generic Galois extension with group G over a field F. Let R be a communtative ring of the form F[x(1),..., x(n)](1/s) and let S be a Galois extension of R with group G. Then S/R is generic for G over F if the following holds. Assume K/L is a Galois extension of fields with group G and such that L superset, dbl equals F. Then there is an F algebra map f:R --> L such that K congruent with S [unk](R)L. We construct generic Galois extensions for certain G and F. We show such extensions are related to Noether's problem and the Grunwald-Wang theorem. One consequence is a simple proof of known counter examples to Noether's problem. On the other hand, we have an elementary proof of a chunk of the Grunwald-Wang theorem, and in a more general context. In fact, we have a Grunwald-Wang-type theorem whenever there is a generic extension for a group G over a field F.
Saltman
D J
DJ
Department of Mathematics, Yale University, New Haven, Connecticut 06520.