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): = x³ - tx² - (t + 3)x - 1 with t ∈ Z. Then we give an equivalent condition for when K_t is monogenic, which is explicitly written in terms of t.