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

For a field k and a finite group G acting regularly on a set of indeterminates (X) under bar = {X(g)}g is an element of G, let k(G) denote the invariant field k((X) under bar)(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, is replaced by an arbitrary finite central extension of either A(n) or S(n), valid over Q(zeta(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))(A5/Q); (2) an affirmative answer to Noether's problem over Q for both (A) over tilde (5) and (S) over tilde (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. (c) 2009 Elsevier Inc. All rights reserved.