On the arithmetic of rational function coefficients with linear-fractional argument changes (Q1358610)

Let \(k\) be a ground field of characteristic \(\not=2\), let \(\overline k\) be an algebraic closure of \(k\), and let \(f\in\overline k(z)\backslash\overline k\) be a nonconstant rational function. The Galois group \(\Gamma=\text{Gal}(\overline k/k)\) and \(G=\text{PSL}_2(\overline k)\) both act on \(\overline k(z)\backslash\overline k\), the latter by fractional linear transformations. Let \([f]\) be the orbit of \(f\) under \(G\) and let \(\Gamma_{[f]}\) be the isotropy subgroup of \([f]\) in \(\Gamma\); the corresponding field is \(k_{[f]}\). This field \(k_{[f]}\) is called the small field of definition of the function \(f\). The author proves that a rational function \(f\in k(z)\backslash k\) can be transformed by a fractional-linear transformation of its argument to have a field of definition that is at most a quadratic extension of the small field of definition of \(f\). The proof is a nice use of Galois cohomology.