A note on monogenic even polynomials

TL;DR

This paper classifies even sextic monogenic polynomials with cyclic Galois groups, proving a conjecture and advancing understanding of their existence, with implications for broader classes of polynomials.

Contribution

It proves a conjecture by Lenny Jones, completing the classification of even sextic monogenic polynomials with cyclic Galois groups.

Findings

Classified even sextic monogenic polynomials with cyclic Galois group

Proved a conjecture of Lenny Jones

Provided insights relevant for general families of even polynomials

Abstract

We extend several predecessor works on even sextic monogenic polynomials. In particular, we prove a conjecture of Lenny Jones, thereby classifying even sextic monogenic polynomials with cyclic Galois group. This result is key to completing previous partial results on existence or non-existence of infinite families of even sextic monogenic polynomials with a prescribed Galois group. Some of the underlying ideas are relevant for investigation of more general families of even polynomials $f(X^2)$, or power-compositional polynomials $f(X^\ell)$.