The plus closure in degree two extensions
Let \(R\) be a local regular ring with quotient field \(K\) and \(R^+\) the integral closure of \(R\) in the algebraic closure of \(K\). Let \(x, y\) be a part of a regular system of parameters and assume that \(p=\) the characteristic of the residue field of \(R\) belongs to \(I= (x,y)R\). Consider \(z\in R^+\) such that its minimal polynomial over \(K\) is \(f(T)= T^2+ c_1 T+ c_2\in R[T]\) and set \(\Delta= c_1^2- 4c_2\). The author gives necessary and sufficient conditions for \(z\) to be in \(IR^+\). For instance, when \(p=3\), he shows that \(z\in IR^+\) iff \(c_1\in I\) and either \(\Delta\in t^2 R\) for some \(t\in I\) or \(\Delta\in (x^3, x^2 y^2, y^3)R\). For this purpose, he develops more general techniques.
scientific article; zbMATH DE number 1348175