A new Proof of Desingularization over fields of characteristic zero

We present a proof of embedded desingularization for closed subschemes whichdoes not make use of Hilbert-Samuel function and avoids Hironaka's notion ofnormal flatness. This proof, already sketched in [A course on constructive desingularizationand equivariance. In {\em Resolution of singularities (Obergurgl, 1997)}, vol.181 {\em Progr. Math.}, Birkh\"auser, 2000.] page 224, is done by showing thatdesingularization of a closed subscheme $X$, in a smooth sheme W, is achievedby taking an algorithmic principalization for the ideal $I(X)$, associated tothe embedded scheme $X$.