## Lhoussain El Fadil

*On prolongations of rank one discrete valuations*

Comment.Math.Univ.Carolin. 60,3 (2019) 299-304.**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).

**Keywords:** discrete valuation; extension of valuation; prime ideal factorization

**DOI:** DOI 10.14712/1213-7243.2019.017

**AMS Subject Classification:** 13A18 11S05

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