PRIME IDEAL FACTORIZATION IN A NUMBER FIELD VIA NEWTON POLYGONS

Let K be a number field defined by an irreducible polynomial F (X) is an element of Z[X] and Z(K) its ring of integers. For every prime integer p, we give sufficient and necessary conditions on F (X) that guarantee the existence of exactly r prime ideals of Z(K) lying above p, where F (X) factors into powers of r monic irreducible polynomials in F-p[X]. The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly r prime ideals of Z(K) lying above p. We further specify for every prime ideal of Z(K) lying above p, the ramification index, the residue degree, and a p-generator.