  Let $K$ be the number field determined by a monic irreducible polynomial
$f(x)$ with integer coefficients. In previous papers we parameterized the prime
ideals of $K$ in terms of certain invariants attached to Newton polygons of
higher order of the defining equation $f(x)$. In this paper we show how to
carry out the basic operations on fractional ideals of $K$ in terms of these
constructive representations of the prime ideals. From a computational
perspective, these results facilitate the manipulation of fractional ideals of
$K$ avoiding two heavy tasks: the construction of the maximal order of $K$ and
the factorization of the discriminant of $f(x)$. The main computational
ingredient is Montes algorithm, which is an extremely fast procedure to
construct the prime ideals.
