Saturated subfields and invariants of finite groups

TL;DR

This paper investigates the structure of invariant subfields under finite group actions on rational function fields, focusing on the concept of saturation and the role of generative elements in understanding these subfields.

Contribution

It introduces the notion of saturated subfields within invariant fields and explores their properties and significance in the context of finite group automorphisms.

Findings

Characterization of saturated subfields of invariants

Identification of conditions for saturation in invariant subfields

Insights into the structure of generative elements in invariant theory

Abstract

Every subfield $\kk(\phi)$ of the field of rational functions $\kk(x_1,...,x_n)$ is contained in a unique maximal subfield of the form $\kk(\psi)$. The element $\psi$ is called generative for the element $\phi$. A subfield of $\kk(x_1,...,x_n)$ is called saturated if it contains a generative element of each its element. We study the saturation property for subfields of invariants $\kk(x_1,...,x_n)^G$, where $G$ is a finite group of automorphisms of the field $\kk(x_1,...,x_n)$.