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 , let k(G) denote the invariant field . We first prove for the alternating group An that, if n is odd, then Q(An) is rational over Q(An−1). We then obtain an analogous result where An is replaced by an arbitrary finite central extension of either An or Sn, valid over Q(ζN) for suitable N. Concrete applications of our results yield: (1) a new proof of Maeda's result on the rationality of QA5(X1,…,X5)/Q; (2) an affirmative answer to Noether's problem over Q for both and ; (3) an affirmative answer to Noether's problem over C for every finite central extension group of either An or Sn with n⩽5.