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. 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 $t$.