Twisted multiplicative field invariants, Noether's problem, and Galois extensions (Q914754)

The author considers multiplicative invariant fields \(F(Q)^ G\) (G finite group, Q a G-lattice; F(Q) the fraction field of the group algebra F[G], where F(Q) is provided with the natural G-action). For certain lattices Q, \(F(Q)^ G\) is stably isomorphic to \(F(G')\) with \(G'\) a split extension of G with an abelian kernel. The main part of this paper is devoted to showing that \(F(G')\) can be considered as an \(\alpha\)-twisted multiplicative invariant field of G. This work is a natural continuation of the author's previous work. E.g. \(\alpha\)-twisted groups can be considered as a way of describing the invariant fields of reductive algebraic groups. The embedding problem is related to \(\alpha\)-twisted multiplicative invariant fields. Unirationality of certain \(\alpha\)-twisted multiplicative invariant fields is shown to be equivalent to the existence of solutions for the embedding problem, thus extending the existing number-theoretic results.