Rationality problem for transitive subgroups of S_8

TL;DR

This paper investigates the rationality of fixed fields under the action of transitive subgroups of S_8, establishing conditions under which these fields are rational, with special focus on characteristic 2 cases.

Contribution

It proves that for most transitive subgroups of S_8, the fixed field is rational, except for specific groups like A_8 and PGL(2,7), especially addressing characteristic 2 scenarios.

Findings

Fixed fields are rational for all transitive subgroups except A_8 and PGL(2,7).

Special considerations are given to characteristic 2 cases.

Provides conditions under which the rationality holds.

Abstract

For any field K and any transitive subgroup G of S_8, let G acts naturally on K(x_1, . . ., x_8) by permutations of the variables, we prove that under some minor conditions K(x_1, . . ., x_8)^G is always K-rational except G is A_8 or G is isomorphic to PGL(2, 7). We pay special attentions on the characteristic 2 cases.