The characterization of cyclic cubic fields with power integral bases

TL;DR

This paper characterizes when cyclic cubic fields are monogenic, showing they are related to simplest cubic fields generated by Shanks polynomials, and provides explicit conditions for monogenity based on the parameter t.

Contribution

It establishes an equivalent condition for the monogenity of cyclic cubic fields and relates it to simplest cubic fields generated by Shanks polynomials, with explicit criteria.

Findings

Monogenic cyclic cubic fields are exactly the simplest cubic fields $K_t$.

Explicit conditions for monogenity are given in terms of the parameter t.

Provides a new characterization linking monogenity to specific cubic polynomials.

Abstract

We provide an equivalent condition for the monogenity of the ring of integers of any cyclic cubic field. We show that if a cyclic cubic field is monogenic then it is a simplest cubic field $K_t$ which is the splitting field of a Shanks cubic polynomial $f_t(x):=x^3-tx^2-(t + 3)x-1$ with $t \in \mathbb Z$. Moreover we give an equivalent condition for when $K_t$ is monogenic, which is explicitly written in terms of $t$.