The characterization of cyclic cubic fields with power integral bases

2019/12/06 by Tomokazu Kashio, Kashio, Tomokazu, Ryutaro Sekigawa +1 · 1 citation
Mathematics · #11C08 #11D25 #11D57 #11N32 #11R04 #11R09 #11R16 #11R27 #11R29 #11R34 #11R80 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1912.03103

openalex publication_date 2019/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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

Citations

• Non monogénéité de l'anneau des entiers des extensions cycliques de Q de degré premier l ≥ 5

Cited by

• Well-Rounded Twists of the Ring of Integers in Cyclic Cubic Fields

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