Monogenic Even Cyclic Sextic Polynomials

TL;DR

This paper investigates the existence of monogenic properties in even cyclic sextic polynomials, proving non-existence for certain types and identifying specific monogenic cases among quadrinomials with cyclic Galois groups.

Contribution

It establishes the non-existence of monogenic even cyclic sextic binomials and trinomials, and characterizes four monogenic quadrinomials within particular infinite sets.

Findings

No monogenic even cyclic sextic binomials or trinomials exist.

Exactly four monogenic quadrinomials are found within certain infinite sets.

The Galois group structure significantly affects monogenic properties.

Abstract

Suppose that $f(x)\in {\mathbb Z}[x]$ is monic and irreducible over ${\mathbb Q}$ of degree $N$. We say that $f(x)$ is monogenic if $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$, and we say $f(x)$ is cyclic if the Galois group of $f(x)$ over ${\mathbb Q}$ is isomorphic to the cyclic group of order $N$. In this note, we prove that there do not exist any monogenic even cyclic sextic binomials or trinomials. Although the complete story on monogenic even cyclic sextic quadrinomials remains somewhat of a mystery, we nevertheless determine that the union of three particular infinite sets of cyclic sextic quadrinomials contains exactly four quadrinomials that are monogenic with distinct splitting fields. We also show that the situation can be quite different for quadrinomials whose Galois group is not cyclic.