Frobenius groups and retract rationality
Journal
Advances in Mathematics
Journal Volume
245
Pages
34-51
Date Issued
2013
Author(s)
Abstract
Let k be any field, G be a finite group acting on the rational function field k (x g: g ∈ G) by h {dot operator} x g = x h g for any h, g ∈ G. Define k(G)=k(xg:g∈G)G. Noether's problem asks whether k (G) is rational (= purely transcendental) over k. A weaker notion, retract rationality introduced by Saltman, is also very useful for the study of Noether's problem. We prove that, if G is a Frobenius group with abelian Frobenius kernel, then k (G) is retract k-rational for any field k satisfying some mild conditions. As an application, we show that, for any algebraic number field k, for any Frobenius group G with Frobenius complement isomorphic to SL2(F5), there is a Galois extension field K over k whose Galois group is isomorphic to G, i.e. the inverse Galois problem is valid for the pair (G, k) The same result is true for any non-solvable Frobenius group if k (ζ8) is a cyclic extension of k. © 2013 Elsevier Ltd.
Type
journal article