On Noether's problem for central extensions of symmetric and alternating groups

Abstract

For a field k and a finite group G acting regularly on a set of indeterminates X ̲ = { X g } g ∈ G , let k ( G ) denote the invariant field k ( X ̲ ) G . We first prove for the alternating group A n that, if n is odd, then Q ( A n ) is rational over Q ( A n − 1 ) . We then obtain an analogous result where A n is replaced by an arbitrary finite central extension of either A n or S n , valid over Q ( ζ N ) for suitable N. Concrete applications of our results yield: (1) a new proof of Maeda's result on the rationality of Q ( X 1 , … , X 5 ) A 5 / Q ; (2) an affirmative answer to Noether's problem over Q for both A 5 ˜ and S 5 ˜ ; (3) an affirmative answer to Noether's problem over C for every finite central extension group of either A n or S n with n ⩽ 5 .