On the integrality of \'etale extensions of polynomial rings

We use a valuation theorem for Noetherian rings, recently obtained by Rangachev, to prove that for any non-integral \'etale extension A⊂ B of polynomial rings over an algebraically closed field of characteristic zero, the integral closure Ā of A in B has infinite cyclic divisor class group, where the generator is a prime divisor equal to the complement of Spec(B) in Spec(Ā). Moreover, this prime divisor coincides with the ramification divisor of the finite extension A⊂ Ā. We apply this result to carry out Wright's geometric approach for two-dimensional non-integral \'etale extensions. It follows from our main theorem and the work of Miyanishi that Spec(Ā) is a smooth affine surface. We show that Spec(Ā) is an A¹-bundle over P¹, more precisely a Danilov-Gizatullin surface of index three. Based on Wright's analysis of which of these affine surfaces can factorize an \'etale morphism of the complex affine plane and his description of its affine coordinate rings we prove that any two-dimensional complex \'etale extension is integral.