ON VALUATIONS OF K(X)

For a valued field (K,v), let K(v) denote the residue field of v and G(v) its value group. One way of extending a valuation v defined on a field K to a simple transcendental extension K(x) is to choose any alpha in K and any mu in a totally ordered Abelian group containing G(v) and define a valuation w on K[x] by w(SIGMA(i)c(i)(x-alpha)i) = min(i) (v(c(i))+i mu). Clearly either G(v) is a subgroup of finite index in G(w)=G(v)+Zmu or G(w)/G(v) is not a torsion group. It can be easily shown that K(x)w is a simple transcendental extension of K(v) in the former case. Conversely it is well known that for an algebraically closed field K with a valuation v, if w is an extension of v to K(x) such that either K(x)w is not algebraic over K(v) or G(w)/G(v) is not a torsion group, then w is of the type described above. The present paper deals with the converse problem for any field K. It determines explicitly all such valuations w together with their residue fields and value groups.