On a problem of Pethő

In this paper we deal with a problem of Pethő related to existence of a quartic algebraic integer α for whichβ=4α4α4−1−αα−1 is a quadratic algebraic number. By studying rational solutions of certain Diophantine system we prove that there are infinitely many α's such that the corresponding β is quadratic. Moreover, we present a description of quartic numbers α such that the corresponding β is a quadratic real number.