On a conjecture concerning the minimal index of pure quartic fields

Monogeneous pure quartic fields $\mathbb{Q}(\sqrt[4]{m})$ are not completely described, not even if $m$ is square-free. I. Gaál and L. Remete [7] formulated a conjecture stating that there are only two monogeneous pure quartic fields with square-free $m$ satisfying $m\equiv 9 \pmod{16}$. We disprove it by showing the existence of infinitely many monogeneous fields of this type if the $abc$ conjecture is true. In this paper, we study the minimal index of pure quartic fields and find all elements with minimal index in totally complex pure quartic fields having a square-free parameter $m$.