Dat T. Tran ; Nam H. Le ; Ha T. N. Tran - Well-Rounded ideal lattices of cyclic cubic and quartic fields

cm:11138 - Communications in Mathematics, October 18, 2023, Volume 31 (2023), Issue 2 (Special issue: Euclidean lattices: theory and applications) - https://doi.org/10.46298/cm.11138
Well-Rounded ideal lattices of cyclic cubic and quartic fieldsArticle

Authors: Dat T. Tran ; Nam H. Le ; Ha T. N. Tran

In this paper, we find criteria for when cyclic cubic and cyclic quartic fields have well-rounded ideal lattices. We show that every cyclic cubic field has at least one well-rounded ideal. We also prove that there exist families of cyclic quartic fields which have well-rounded ideals and explicitly construct their minimal bases. In addition, for a given prime number $p$, if a cyclic quartic field has a unique prime ideal above $p$, then we provide the necessary and sufficient conditions for that ideal to be well-rounded. Moreover, in cyclic quartic fields, we provide the prime decomposition of all odd prime numbers and construct an explicit integral basis for every prime ideal.

Comment: 42 pages

https://doi.org/10.46298/cm.11138
Source: arXiv.org:2303.16968
Volume: Volume 31 (2023), Issue 2 (Special issue: Euclidean lattices: theory and applications)
Published on: October 18, 2023
Accepted on: August 28, 2023
Submitted on: March 31, 2023
Keywords: Mathematics - Number Theory, 11R16, 06B10, 06B99, 11Y40
Licence: Attribution 4.0 International (CC BY 4.0)
Funding:
    Source : OpenAIRE Graph
  • Funder: Natural Sciences and Engineering Research Council of Canada

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