Quartic Index Form Equations and Monogenizations of Quartic Orders

TL;DR

This paper investigates the number of ways quartic orders can be generated by a single element, using classical Diophantine equations like index form and Thue equations to establish upper bounds.

Contribution

It introduces new upper bounds for monogenizations of quartic orders through analysis of index form and Thue equations in quartic fields.

Findings

Established upper bounds for monogenizations of quartic orders.

Applied classical Diophantine equations to bound the number of generators.

Connected index form equations with monogenization properties.

Abstract

Some upper bounds for the number of monogenizations of quartic orders are established by considering certain classical Diophantine equations, namely index form equations in quartic number fields, and cubic and quartic Thue equations.