  We adapt an old local-to-global technique of Ore to compute, under certain
mild assumptions, an integral basis of a number field without a previous
factorization of the discriminant of the defining polynomial. In a first phase,
the method yields as a by-product successive splittings of the discriminant.
When this phase concludes, it requires a squarefree factorization of some base
factors of the discriminant to terminate.
