Frobenius groups and retract rationality

TL;DR

This paper proves that for Frobenius groups with abelian kernels, the associated fixed field is retract rational over any field meeting certain conditions, impacting the inverse Galois problem.

Contribution

It establishes retract rationality of fixed fields for a class of Frobenius groups, extending understanding of Noether's problem and Galois realizations.

Findings

Proves retract rationality for Frobenius groups with abelian kernels.

Shows the inverse Galois problem holds for certain Frobenius groups over algebraic number fields.

Extends results to non-solvable Frobenius groups under specific field extension conditions.

Abstract

Let $k$ be any field, $G$ be a finite group acting on the rational function field $k(x_g:g\in G)$ by $h\cdot x_g=x_{hg}$ for any $h,g\in G$. Define $k(G)=k(x_g:g\in G)^G$. 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 $SL_2(\bm{F}_5)$, 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 $k(\zeta_8)$ is a cyclic extension of $k$.