Monogenic Cyclic Cubic Trinomials

TL;DR

This paper classifies monogenic cyclic cubic trinomials over integers, identifying two infinite families of the form x^3+Ax+B and exactly four of the form x^3+Ax^2+B, with one known as x^3-3x+1.

Contribution

It extends the classification of monogenic cyclic polynomials from quartic to cubic cases, discovering new infinite families and specific known examples.

Findings

Two infinite families of monogenic cyclic cubic trinomials of the form x^3+Ax+B.

Exactly four monogenic cyclic cubic trinomials of the form x^3+Ax^2+B, all equivalent to x^3-3x+1.

Complete classification of monogenic cyclic cubic trinomials in the studied forms.

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 $x^4+bx^2+d$. 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 $x^3-3x+1$.