The characterization of cyclic cubic fields with power integral bases

• Kashio Tomokazu

  Department of Mathematics Faculty of Science and Technology Tokyo University of Science

• Sekigawa Ryutaro

  Department of Mathematics Faculty of Science and Technology Tokyo University of Science

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.

Journal

• Kodai Mathematical Journal

  Kodai Mathematical Journal 44 (2), 290-306, 2021-06-30

  Department of Mathematics, Tokyo Institute of Technology

Keywords

• monogenity
• power integral basis
• cyclic cubic fields
• Shanks cubic polynomials
• simplest cubic fields
• ambiguous ideals
• Galois cohomology
• unit group
• Newton polygon

Details

• CRID: 1390570010598130560
• NII Article ID: 130008059294
• DOI: 10.2996/kmj44204
• ISSN: 18815472
• ISSN: 03865991
• Text Lang: en
• Data Source:
  • JaLC
  • Crossref
  • CiNii Articles
  • KAKEN
• Abstract License Flag: Disallowed