Finite solvable groups with a rational skew-field of noncommutative real rational invariants

TL;DR

This paper investigates the rationality of invariant skew-fields under finite group actions, establishing conditions for solvable groups and classifying certain p-groups with specific properties.

Contribution

It introduces the concept of totally pseudo-unramified groups and characterizes when invariant skew-fields are rational for solvable groups.

Findings

Invariant skew-fields are always rational for abelian groups.

Invariant skew-fields are finitely generated for solvable groups.

Classification of totally pseudo-unramified p-groups of rank ≤ 5.

Abstract

We consider the Noether's problem on the noncommutative real rational functions invariant under the linear action of a finite group. For abelian groups the invariant skew-fields are always rational. We show that for a solvable group the invariant skew-field is finitely generated. The skew-field invariant under a linear action of a solvable group is rational if the action is well-behaved -- given by a so-called complete representation. We determine the groups that admit such representations and call them totally psuedo-unramified. In the second part we study the reach of totally psuedo-unramified groups and classify totally pseudo-unramified p-groups of rank at most 5.