The invertibility of an element \(\alpha^2-a\) of a super-primitive extension \(R[\alpha]/R\) and a linear form of a Laurent extension \(R[\alpha, \alpha^{-1}]\) (Q2707046)

Let \(R\) be an integral domain with quotient field \(K\). Let \(\alpha\) be algebraic over \(K\) with minimal polynomial \(\phi(x) = x^d + \eta_1 x^{d-1} + \ldots + \eta_d\), let \(I_{[\alpha]}\) be the ideal of elements of \(R\) that are common denominators of the coefficients of \(\phi(x)\), and let \(J_{[\alpha]}\) be the ideal \(I_{[\alpha]}\cdot (1, \eta_1, \ldots, \eta_d)\). If \(J_{[\alpha]}\) is not contained in any prime ideal \(p\) of \(R\) with depth\((R_p) = 1\), then \(\alpha\) is called super-primitive over \(R\). A super-primitive element is an anti-integral element, by a result of \textit{S. Oda, J. Sato} and \textit{K.-I. Yoshida} [Osaka J. Math. 30, 119-135 (1993; Zbl 0782.13013)]. For \(\alpha\) super-primitive over \(R\), the authors relate the invertibility of elements \(\alpha^2 - c\) in \(R[\alpha]\) and of \(c\alpha - d\) in \(R[\alpha, \alpha^{-1}]\) for \(c, d\) in \(R\) to properties of the radical of the \(\mathbb Z\)-module \(Rc\) and other \(\mathbb Z\)-module radicals.