On a problem of Pethő

Abstract

In this paper we deal with a problem of Pethő related to existence of quartic algebraic integer $\alpha$ for which $$\beta =\frac{4{\alpha }^4}{{\alpha }^4-1}-\frac{\alpha }{\alpha -1}$$ is a quadratic algebraic number. By studying rational solutions of certain Diophantine system we prove that there are infinitely many $\alpha$'s such that the corresponding $\beta$ is quadratic. Moreover, we present a description of all quartic numbers $\alpha$ such that $\beta$ is quadratic real number.