Let A be a Dedekind domain whose field of fractions K is a global field. Let p be a non-zero prime ideal of A, and Kp the completion of K atp. The Montes algorithm factorizes a monic irreducible separable polynomial f(x) ∈ A[x] over Kp, and it provides essential arithmetic information about the finite extensions of Kp determined by the different irreducible factors. In particular, it can be used to compute a p-integral basis of the extension of K determined by f(x). In this paper we present a new and faster method to compute p-integral bases, based on the use of the quotients of certain divisions with remainder of f(x) that occur along the flow of the Montes algorithm