Abstract. Let $\mathrm{Aut}_\K \K(x)$ be the Galois group of the transcendental degree one pure field extension $\K\subseteq\K(x)$. In this paper we describe polynomial time algorithms for computing the field $Fix(H)$ fixed by a subgroup $H$ of $\mathrm{Aut}_\K \K(x)$ and for computing the fixing group $G_f$ of a rational function $f \in \K(x)$.