Abstract
Let K be a number field defined by a monic irreducible polynomial F (X) is an element of Z [X], p a fixed rational prime, and nu(p) the discrete valuation associated to p. Assume that (F) over bar (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. (C) 2019 Mathematical Institute Slovak Academy of Sciences