Three-dimensional purely monomial group actions (Q1340979)

A \(K\)-automorphism \(\sigma\) of the field \(K(x_1, \ldots, x_n)\) is called monomial if \(\sigma (x_i) = a_i (\sigma) \prod^n_{j = 1} x_j^{m_{ij}}\), where \((m_{ij})_{1 \leq i,j \leq n}\) is an invertible \(n \times n\) matrix with integer entries and \(a_i (\sigma) \in K \backslash \{0\}\). If \(a_i (\sigma) = 1\), then the \(K\)- automorphism \(\sigma\) is called purely monomial. The main result is contained in the following theorem: For any field \(K\), the fixed point set of \(K(x_1, x_2, x_3)\) under any purely monomial group action is rational over \(K\) except for those which are conjugate in \(GL (3,\mathbb{Z})\) to the subgroup generated by \[ \left( \begin{matrix} 1 & 1 & 0 \\ -2 & -1 & -1 \\ 0 & 0 & 1 \end{matrix} \right) \quad \text{and} \quad \left( \begin{matrix} -1 & -1 & -1 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} \right). \]