The characterization of cyclic cubic fields with power integral bases

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 Kₜ which is the splitting field of a Shanks cubic polynomial fₜ(x):=x³-tx²-(t + 3)x-1 with t ∈ Z. Moreover we give an equivalent condition for when Kₜ is monogenic, which is explicitly written in terms of t.