Some group actions on \(K(x_1,x_2,x_3)\)
The main object of the paper under review is \(F=K(x_1,x_2,x_3)^{\left<\sigma\right>}\), the field of invariants of a purely transcendental extension \(K(x_1,x_2,x_3)\) of degree 3 of an arbitrary field \(K\) with respect to a fractional-linear action of a \(K\)-automorphism \(\sigma\): \(x_i \mapsto (a_ix_i+b_i)/(c_ix_i+d_i)\), where \(a_id_i-b_ic_i\neq 0\), \(i=1,2,3\). The author's goal is to provide a criterion for the extension \(F/K\) to be rational (= purely transcendental) in terms of the ``characteristic polynomials'' \(f_i(T)=T^2-(a_i+d_i)T+(a_id_i-b_ic_i)\) of the matrices \(\sigma_i=\left(\begin{smallmatrix} a_i & b_i \\ c_i & d_i \end{smallmatrix}\right)\). The main results are the following. If \(\text{char}(K)\neq 2\), then \(F/K\) is rational if and only if (i) for each \(1\leq i\leq 3\), \(f_i(T)\) is irreducible; (ii) the Galois group of \(f_1(T)f_2(T)f_3(T)\) over \(K\) is of order 8; (iii) for each \(1\leq i\leq 3\), the order of \(\sigma_i\) in \(\text{PGL}_2(K)\) is an even integer. Moreover, if \(F/K\) is not rational, it is not even retract rational. If \(\text{char}(K)=2\), then \(F/K\) is always rational except possibly the case where, for each \(1\leq i\leq 3\), \(f_i(T)\) is irreducible and is of the form \(T^2-a_i\). These results can be viewed as a generalization of [\textit{D. J. Saltman}, A nonrational field, answering a question of Hajja. Algebra and number theory. Proceedings of a conference, Fez, Morocco. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 208, 263--271 (2000; Zbl 0977.12005)]. The proof is based on reducing to the case where \(F\) can be represented in the form \(L(z_1,z_2,z_3)^G\), where \(G\), the Galois group of the splitting field \(L\) of the polynomial \(f_1(T)f_2(T)f_3(T)\) over \(K\), acts linearly on the \(z_i\)'s. In that case \(F\) is isomorphic to \(K(S)\), the field of rational functions of a certain algebraic \(K\)-torus \(S\) split over \(L\), and the author can use birational classification of algebraic tori of small dimensions, in particular, results of the reviewer [Sel. Math. Sov. 9, 1--21 (1990; Zbl 0707.14036)].
scientific article; zbMATH DE number 2174703
scientific article