  Let $A$ be a Dedekind domain, $K$ the fraction field of $A$, and $f\in A[x]$
a monic irreducible separable polynomial. For a given non-zero prime ideal
$\mathfrak{p}$ of $A$ we present in this paper a new method to compute a
$\mathfrak{p}$-integral basis of the extension of $K$ determined by $f$. Our
method is based on the use of simple multipliers that can be constructed with
the data that occurs along the flow of the Montes Algorithm. Our construction
of a $\mathfrak{p}$-integral basis is significantly faster than the similar
approach from $[7]$ and provides in many cases a priori a triangular basis.
