On the extension of a valuation on a field \(K\) to \(K(X)\). II (Q1356817)

Two types of composite valuations of a simple transcendental extension \(K(X)\) of a valued field \((K,v)\) are characterized. Namely, the residually algebraic extension of the first kind and the composite of two residually transcendental extensions are given in theorems 2.1 and 3.1 respectively. Here a valuation \(u\) of \(K(X)\) is said to be of the first kind if there exists a residual transcendental extension \(w\) of \(v\) to \(K(X)\) such that \(u\leq w\). These results are based on the ``fundamental theorem'' of the characterization of residual transcendental extensions of a valuation. As an application one can describe all the valuations of the field of rational functions in \(n\) independent variables over the valued field \((K,v)\). In this connection see part I [\textit{N. Popescu} and \textit{C. Vraciu}, ibid. 87, 151-168 (1992; Zbl 0774.12009)].