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 $x^4-5x^2+5$.
Categories: math.NT