Generic Galois extensions and problems in field theory

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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 Fx(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. (1982). Generic Galois extensions and problems in field theory. Advances in Mathematics, 43(3), 250–283.

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