A two-dimensional rationality problem and intersections of two quadrics

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Let k be a field with char \(k\ne 2\) and k be not algebraically closed. Let \(a\in k{\setminus } k^2\) and \(L=k(\sqrt{a})(x,y)\) be a field extension of k where x, y are algebraically independent over k. Assume that \(\sigma \) is a k-automorphism on L defined by $$\begin{aligned} \sigma : \sqrt{a}\mapsto -\sqrt{a},\ x\mapsto \frac{b}{x},\ y\mapsto \frac{c\big (x+\frac{b}{x}\big )+d}{y} \end{aligned}$$where \(b,c,d \in k\), \(b\ne 0\) and at least one of c, d is non-zero. Let \(L^{\langle \sigma \rangle }=\{u\in L:\sigma (u)=u\}\) be the fixed subfield of L. We show that \(L^{\langle \sigma \rangle }\) is isomorphic to the function field of a certain surface in \(\mathbb {P}^4_k\) which is given as the intersection of two quadrics. We give criteria for the k-rationality of \(L^{\langle \sigma \rangle }\) by using the Hilbert symbol. As an appendix of the paper, we also give an alternative geometric proof of a part of the result which is provided to the authors by J.-L. Colliot-Thélène.