A family of cyclic quartic monogenic polynomials

TL;DR

This paper introduces explicit families of cyclic quartic polynomials that are often monogenic and, under the abc conjecture, generate infinitely many distinct monogenic quartic fields, expanding the understanding of such number fields.

Contribution

It provides explicit constructions of cyclic quartic polynomials that are monogenic and explores their infinite families under the abc conjecture.

Findings

Explicit families of cyclic quartic monogenic polynomials are constructed.

Under the abc conjecture, these families generate infinitely many distinct monogenic quartic fields.

Additional conjectural families are proposed, suggesting a rich structure of such fields.

Abstract

We produce an explicit family of totally real cyclic quartic polynomials that are monogenic in many cases and, if the $abc$ conjecture holds, generate distinct monogenic quartic fields infinitely often. Additional families (also conjecturally generating infinitely many distinct fields) are provided in Section 4, including what appears to be an infinite collection of such families.