Transcendental Galois Theory and Noether's Problem

Abstract

In 1918, Noether published a paper where she studied such a problem, now called Noether's problem on rationality: Let $L=K\left(t_1,t_2,\cdots ,t_n\right)$ be a purely transcendental extension over a field $K$ and $G$ a finite subgroup acting transitively on $t_1,t_2,\cdots ,t_n$ in an evident manner. Is it true that the invariant subfield $L^G$ of $L$ under is still purely transcendental over $K$? The problem has been open in general except for minor particular cases. In this paper we will attempt to understand a general theory for Noether's problem on rationality by transcendental Galois theory. Then new particular cases will be obtained. We will also give a generalization for the remarkable counter-example given by Swan in 1969.

111 pages. Comments welcome

References in corpus (1)

• On the transcendental Galois extensions