Let \(K\) be a field, \(\sigma\) an automorphism of \(K\) and \(v\) a \(\sigma\)-invariant valuation on \(K\). The main goal of the paper under review is to study all the extensions of \(v\) to the skew field \(K(X,\sigma)\) which consists of the left quotients of the skew polynomial ring \(K[X,\sigma]\) where \(Xa=\sigma(a)X\) for any \(a\in K\). When \(\sigma\) is the identity, \(K(X,\sigma)= K(X)\) and the problem was solved by \textit{V. Alexandru, N. Popescu} and \textit{A. Zaharescu} [J. Math. Kyoto Univ. 30, No. 2, 281--296 (1990; Zbl 0728.12010)]. The author studies the case when \(K\) is algebraically closed. In this situation, if \(\sigma\) is not the identity, its order is two. He restricts to the case that the residue field is of characteristic different from two. The main result establishes that any residually transcendental extension \(w\) of \(v\) to \(K(X,\sigma)\) is of degree one. Residually transcendental means that there exists an element in the valuation ring such that its residue class is transcendental over the valuation ring of \(v\).