Calculating the index of equation orders using Newton polygons

Montes and Nart made an important result by carefully addressing the index of orders and extending Dedekind's and Ore's results on the correspondence between the factorization of prime ideals and polynomials. They further gave a necessary and sufficient condition for the applicability of their theorem. Unfortunately, they omitted the proof, and there was a minor gap in the condition. The aim of this paper is to state their theorem in a precise form, provide a complete proof, and present applications of the theorem.