The characterization of cyclic cubic fields with power integral bases

Abstract

We provide an equivalent condition for the monogenity of the ring of integers of any cyclic cubic field. Although a large part of the main results is covered by the classical one of Gras, we write the condition more explicitly. First, we show that if a cyclic cubic field is monogenic then it is a simplest cubic field $K_t$ defined by Shanks' cubic polynomial $f_t(x):=x^3-tx^2-(t + 3)x-1$ with $t \in \mathbf{Z}$. Then we give an equivalent condition for when $K_t$ is monogenic, which is explicitly written in terms of $t$.