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 \(x^4+bx^2+d\) . 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 \(x^3+Ax^2+B\) , all of which are equivalent to \(x^3-3x+1\) .
