  Let $K$ be a number field defined by a monic irreducible polynomial $F(X) \in
\mathbb{Z}[X]$, $p$ a fixed rational prime, and $\nu_p$ the discrete valuation
associated to $p$. Assume that $\overline{F}(X)$ factors modulo $p$ into the
product of powers of $r$ distinct monic irreducible polynomials. We present in
this paper a condition, weaker than the known ones, which guarantees the
existence of exactly $r$ valuations of $K$ extending $\nu_p$. We further
specify the ramification indices and residue degrees of these extended
valuations in such a way that generalizes the known estimates. Some useful
remarks and computational examples are also given to highlight some
improvements due to our result.
