Monogenic Cyclic Cubic Trinomials

Abstract

A series of recent articles has shown that there exist only three monogenic cyclic quartic trinomials in $\mathbb{Z}[x]$, and they are all of the form $x^4+b{x}^2+d$. In this article, we conduct an analogous investigation for cubic trinomials in $\mathbb{Z}[x]$. Two irreducible cyclic cubic trinomials are said to be equivalent if their splitting fields are equal. We show that there exist two infinite families of non-equivalent monogenic cyclic cubic trinomials of the form $x^3+Ax+B$. We also show that there exist exactly four monogenic cyclic cubic trinomials of the form $x^3+A{x}^2+B$, all of which are equivalent to $x^3-3x+1$.