Construction of the integral closure of an affine domain in a finite field extension of its quotient field

The construction of the normalization of an affine domain over a field is a classical problem solved since sixteenʹs by Stolzenberg (1968) and Seidenberg (1970–1975) thanks to classical algebraic methods and more recently by Vasconcelos (1991–1998) and de Jong (1998) thanks to homological methods. The aim of this paper is to explain how to use such a construction to obtain effectively the integral closure of such a domain in any finite extension of its quotient field, thanks to Dieudonné characterization of such an integral closure. As application of our construction, we explain how to obtain an effective decomposition of a quasi-finite and dominant morphism from a normal affine irreducible variety to an affine irreducible variety as a product of an open immersion and a finite morphism, conformly to the classical Grothendieckʹs version of Zariskiʹs main theorem.