Rationality for subgroups of S_6

TL;DR

This paper proves that for certain subgroups of S_6 containing C_3 x C_3, the fixed field of rational functions is rational over any base field, with applications to constructing generic polynomials.

Contribution

It establishes the rationality of fixed fields for specific subgroups of S_6 containing C_3 x C_3, extending known results and providing new tools for generic polynomial construction.

Findings

Fixed fields are rational over any base field for these subgroups.

Provides a method for constructing generic polynomials.

Extends the class of subgroups with known rationality results.

Abstract

For a transitive subgroup $G \le S_6$ which contain $C_3 \times C_3$ as subgroup, we prove that $K(x_1,\dots,x_6)^G$ is rational over $K$, where $K$ is any field, and $G$ acts naturally on $K(x_1,\dots,x_6)$ by permutations on the variables. We also give an application on construction of generic polynomials.