Monogenic cyclic trinomials of the form $x^4+cx+d$

TL;DR

This paper proves that no monogenic cyclic quartic trinomials of the form $x^4+cx+d$ exist, and identifies all monogenic cyclic quartic trinomials, clarifying the structure of such polynomials.

Contribution

It establishes the non-existence of monogenic cyclic quartic trinomials of the form $x^4+cx+d$ and classifies all such polynomials.

Findings

No monogenic cyclic quartic trinomials of the form $x^4+cx+d$ exist.

The only monogenic cyclic quartic trinomials are $x^4-4x^2+2$, $x^4+4x^2+2$, and $x^4-5x^2+5$.

The result completes the classification of monogenic cyclic quartic trinomials.

Abstract

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $n$ that is irreducible over ${\mathbb Q}$ is called cyclic if the Galois group over ${\mathbb Q}$ of $f(x)$ is the cyclic group of order $n$, while $f(x)$ is called monogenic if $\{1,\theta,\theta^2,\ldots, \theta^{n-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. In this article, we show that there do not exist any monogenic cyclic trinomials of the form $f(x)=x^4+cx+d$. This result, combined with previous work, proves that the only monogenic cyclic quartic trinomials are $x^4-4x^2+2$, $x^4+4x^2+2$ and $x^4-5x^2+5$.