On prolongations of valuations via Newton polygons and liftings of polynomials
Let \((K,v)\) be a real valued field and \(K(\theta)\) be a finite separable extension of \(K\) with \(\theta\) integral over \(R_v\), the valuation ring of \(v\).The scope of the paper is the description of all prolongations of \(v\) to \(K(\theta)\). This problem is classical but the method in the paper is new. It involves Newton polygons as well as prolongations of \(v\) to a simple transcendental extension \(K(X)\) of \(K\) whose residue field are transcendental over \(K_v\), the residue field of \(v\). The residue field of such a prolongation \(w\) is a simple transcendental extension \(\bar L(Z)\) of a finite extension \(\bar L\) of \(K_v\). The method in the present paper uses also liftings of polynomials belonging to \(\bar L[Z]\) with respect to \(w\) which yield suitable polynomials over \(K\). The authors end the paper by deducing some classical theorems on the subject.
scientific article; zbMATH DE number 6141852