Twisted symmetric group actions (Q611868)

If \(K\) is any field, and if \(K_n = K(x_1,\cdots,x_n)\) is the field of rational functions over \(K\) in \(n\) indeterminates, then the symmetric and alternating groups \(S_n\) and \(A_n\) act on \(K_n\) as groups of \(K\)-automorphisms in the natural way \(\sigma : x_j \mapsto x_{\sigma (j)}\). It is well known that the subfield \(K_n(S_n)\) of \(K_n\) fixed by (this action of) \(S_n\) is generated by the \(n\) elementary symmetric polynomials and is hence rational (= purely transcendental) over \(K\). On the other hand, the question regarding the rationality of \(K_n(A_n)\) has been answered in the very special cases \(n = 3, 4, 5\) only. The cases \(n = 3, 4\) were treated by \textit{M. Hajja} [Algebras Groups Geom. 6, No. 1, 49--54 (1989; Zbl 0714.12008)] and reproduced in Exercises 3, 4 (pp.~524--525) of \textit{N. Jacobson}'s book [Basic Algebra II, 2nd ed., W. H. Freeman and Co., New York (1989; Zbl 0694.16001)], while the case \(n=5\) is settled by \textit{T. Maeda} [J. Algebra, 125, No. 2, 418--430 (1989; Zbl 0697.12018)]. A twisted action of \(S_n\) on \(K_n\) is obtained by taking a non-zero element \(a \in K\) and letting \(S_n\) act on \(K_n\) by \(\tau : x_j \mapsto a / x_{\tau (j)}\) for every transposition \(\tau\). For \(a = 1\) (or equivalently \(a\) is a square), the rationality of the fixed field \(T_n\) under this twisted action is established by \textit{M. Hajja} and \textit{M-c. Kang} in [J. Algebra, 188, No. 2, 626--646 (1997; Zbl 0988.13007)]. The authors of the paper under review consider the case of arbitrary \(a\) (not necessarily a square) and establish the rationality of \(T_n\) for \(n=3, 4, 5\), leaving the cases \(n > 5\) open. This result also answers the question raised by \textit{M. Hajja} in Theorem 5.13 (p.~147) of [M'hammed Boulagouaz (ed.) et al., Algebra and Number Theory. Proceedings of a conference, Fez, Morocco. Marcel Dekker, New York. Lecture Notes Pure Appl. Math. 208 (2000; Zbl 0958.12003)].