On prolongations of rank one discrete valuations

Abstract

Let $(K,\nu)$ be a valued field, where $\nu$ is a rank one discrete valuation. Let $R$ be its ring of valuation, ${\mathfrak m}$ its maximal ideal, and $L$ an extension of $K$, defined by a monic irreducible polynomial $F(X) \in R[X]$. Assume that $\overline{F}(X)$ factors as a product of $r$ distinct powers of monic irreducible polynomials. In this paper a condition which guarantees the existence of exactly $r$ distinct valuations of $K$ extending $\nu$ is given, in such a way that it generalizes the results given in the paper ``Prolongations of valuations to finite extensions" by S.\,K. Khanduja, M. Kumar (2010).