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- W2594177432 abstract "Let k be any field, G be a finite group acting on the rational function field k ( x g : g ∈ G ) by h ⋅ x g = x h g for any h , g ∈ G . Define k ( G ) = k ( x g : 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 S L 2 ( F 5 ) , 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 ( ζ 8 ) is a cyclic extension of k ." @default.
- W2594177432 created "2017-03-16" @default.
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- W2594177432 date "2013-10-01" @default.
- W2594177432 modified "2023-10-14" @default.
- W2594177432 title "Frobenius groups and retract rationality" @default.
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- W2594177432 doi "https://doi.org/10.1016/j.aim.2013.06.009" @default.
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