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

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.

令k为具有char \（k \ ne 2 \）的字段，并且k不是代数封闭的。令\（a \ in k {\ setminus} k ^ 2 \）和\（L = k（\ sqrt {a}）（x，y）\）是k的场扩展，其中x，  y在代数上独立于ķ。假设\（\ sigma \）是由L定义的L的k自同构

其中\（b，c，d \ in k \），\（b \ ne 0 \）和c，  d中的至少一个非零。令\（L ^ {\ langle \ sigma \ rangle} = \ {u \ in L：\ sigma（u）= u \} \）是L的固定子字段。我们证明\（L ^ {\ langle \ sigma \ rangle} \）与\（\ mathbb {P} ^ 4_k \）中某个曲面的函数场是同构的，该曲面由两个二次曲面的交点给出。我们给出\（L ^ {\ langle \ sigma \ rangle} \）的k理性的标准通过使用希尔伯特符号。作为本文的附录，我们还提供了部分结果的替代几何证明，该证明由J.-L提供给作者。科利奥特·泰勒（Colliot-Thélène）。