Fixed fields of triangular matrix groups (Q855746)

Let \(K\) be a field, \(G\) a finite subgroup of \(\text{GL}_n(K)\) acting on the rational function field \(K(x_1,\ldots, x_n)\) by \(K\)-automorphism in the usual manner; let \(K(x_1,\ldots,x_n)^G\) denote the subfield of functions invariant under the action of \(G.\) It was shown by \textit{T. Miyata} [Nagoya Math. J. 41, 69--73 (1971; Zbl 0211.06801)] that if \(G\) consists of upper triangular matrices, then \(K(x_1,\ldots,x_n)^G\) is a purely transcendental extension of \(K.\) In the present note, using more general results of \textit{H. Ahmad, M. Hajja} and \textit{M. Kang} [J. Algebra 228, 643--658 (2000; Zbl 0993.12003)], it is shown that there actually exists a transcendence basis \(f_1,\ldots, f_n \in K[x_1,\ldots, x_n],\) so that \(K(f_1,\ldots, f_n)=K(x_1,\ldots,x_n)^G.\) Note: The references should give \textit{T. A. Springer}'s book [``Invariant theory'' (1977; Zbl 0346.20020)] as LNM 585 (not as GTM 585).