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 $t$.
Acknowledgment
The authors would like to thank Professors Toru Nakahara, Ryotaro Okazaki and Hiroshi Tsunogai for useful discussions. We also thank Master's student Yudai Tanaka for suggesting to apply some techniques in "genus theory" for this topic.
Citation
Tomokazu Kashio. Ryutaro Sekigawa. "The characterization of cyclic cubic fields with power integral bases." Kodai Math. J. 44 (2) 290 - 306, June 2021. https://doi.org/10.2996/kmj44204
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