Transcendental Galois Theory and Noether's Problem

2019/09/02 by Feng-Wen An, An, Feng-Wen
Computer Science · Mathematics · #12F12 #12F20 #19B14 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #History and Theory of Mathematics #K-Theory and Homology (math.KT) #Number Theory (math.NT) #Polynomial and algebraic computation #Primary 14E08 #Secondary 11J81

paper · pdf · doi:10.48550/arxiv.1909.02952

openalex publication_date 2019/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In 1918, Noether published a paper where she studied such a problem, now called Noether's problem on rationality: Let L=K( t₁,t₂,⋯ ,t_n) be a purely transcendental extension over a field K and G a finite subgroup acting transitively on t₁,t₂,⋯ ,t_n in an evident manner. Is it true that the invariant subfield L^G of L under % G 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.

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