Monogenic Cyclic Quartic Trinomials

Abstract

A monic polynomial $f(x)\in \mathbb{Z}[x]$ of degree $N$ is called monogenic if $f(x)$ is irreducible over $\mathbb{Q}$ and $\{1,\theta ,{\theta }^2,\dots ,{\theta }^{N-1}\}$ is a basis for the ring of integers of $\mathbb{Q}(\theta )$, where $f(\theta )=0$. In this brief note, we prove that there exist exactly three distinct monogenic trinomials of the form $x^4+b{x}^2+d$ whose Galois group is the cyclic group of order 4.