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

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 $\{1,\theta ,{\theta }^2,\dots ,{\theta }^{n-1}\}$ 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 $x^4-5x^2+5$.