Comments On " Orbits of automorphism groups of fields"

TL;DR

This paper investigates the structure of noetherian integral domains under automorphism group actions, proving they are fields when the orbit space is finite, and explores conditions under which field extensions are algebraically closed.

Contribution

It establishes that if the automorphism orbit space of a noetherian integral domain is finite, then the domain is a field, and confirms a special case of a conjecture regarding algebraically closed fields.

Findings

R is a field when (R-k)/Aut_k(R) is finite

If (K-k)/Aut_k(K) has size 1, then K is algebraically closed

Elementary proof provided for finitely generated extensions over prime fields

Abstract

Let $R$ be a commutative $k-$algebra over a field $k$. Assume $R$ is a noetherian, infinite, integral domain. The group of $k-$automorphisms of $R$,i.e.$Aut_k(R)$ acts in a natural way on $(R-k)$.In the first part of this article, we study the structure of $R$ when the orbit space $(R-k)/Aut_k(R)$ is finite.We note that most of the results, not particularly relevent to fields, in [1,\S 2] hold in this case as well. Moreover, we prove that $R$ is a field. In the second part, we study a special case of the Conjecture 2.1 in [1] : If $K/k$ is a non trivial field extension where $k$ is algebraically closed and $\mid (K-k)/Aut_k(K) \mid = 1$ then $K$ is algebraically closed. In the end, we give an elementary proof of [1,Theorem 1.1] in case $K$ is finitely generated over its prime subfield.