Monogenic cyclic cubic trinomials

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 $$x^3-3x+1$$ .