Sets of fundamental units in real quadratic and pure cubic fields (Q1903501)

Let \(F(X)\) be an even degree polynomial of \(\mathbb{Q}(X)\) with simple roots and let the leading coefficient of \(F(X)\) be a perfect square. Denote by \(U_X\) the group of units of the ring \(\mathbb{Q}[X, \sqrt F]\) and suppose that the rank of \(U_X\) modulo \(\mathbb{Q}^*\) is 1. Denote by \(E(X)= U(X)+ V(X)\sqrt F\) the fundamental unit of the group of units of norm \(\pm 1\). Suppose there are infinitely many integers \(n\) such that \(F(n)\) is a positive non-square integer and such that \(E(n)\) is a unit of the real quadratic field \(\mathbb{Q}(\sqrt n)\). In the paper under review, the author gives conditions under which the unit \(E(n)\) is the fundamental unit of the real quadratic field \(\mathbb{Q}(\sqrt{F(n)})\). The results obtained are interesting and not trivial at all. The author deals also with the same problem for polynomials of degree \(3m\) and associated pure cubic fields.