Let
$A$ be a Dedekind
domain,
$K$ the
fraction field of
$A$,
and
$f\in A\left[x\right]$
a monic irreducible separable polynomial. For a given nonzero prime ideal
$\mathfrak{p}$ of
$A$
we present in this paper a new algorithm to compute a triangular
$\mathfrak{p}$integral basis of
the extension
$L$
of
$K$ determined
by
$f$.
This approach can be easily adapted to compute a triangular
$\mathfrak{p}$integral basis of
fractional ideals
$I$ of the
integral closure of
$A$ in
$L$. Along this process one can
compute
$\mathfrak{p}$integral bases for a
family of ideals contained in
$I$
as a byproduct.
