  This paper deals with the Peskine version of Zariski Main Theorem published
in 1965 and discusses some applications. It is written in the style of Bishop's
constructive mathematics. Being constructive, each proof in this paper can be
interpreted as an algorithm for constructing explicitly the conclusion from the
hypothesis. The main non-constructive argument in the proof of Peskine is the
use of minimal prime ideals. Essentially we substitute this point by two
dynamical arguments; one about gcd's, using subresultants, and another using
our notion of strong transcendence. In particular we obtain algorithmic
versions for the Multivariate Hensel Lemma and the structure theorem of
quasi-finite algebras.
