# On a problem of Peth\H{o}

## TL;DR

This paper investigates a specific algebraic number problem related to quartic algebraic integers and their associated quadratic algebraic numbers, proving the infinitude of such cases and characterizing all solutions.

## Contribution

It introduces new results on the existence and classification of quartic algebraic integers linked to quadratic algebraic numbers, expanding understanding of this algebraic problem.

## Key findings

- Infinitely many quartic algebraic integers produce quadratic algebraic numbers.
- Complete description of quartic numbers leading to quadratic algebraic numbers.
- Analysis of rational solutions to related Diophantine systems.

## Abstract

In this paper we deal with a problem of Peth\H{o} 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.