Unit Reducible Cyclotomic Fields

2023/11/28 by Chris L. Porter, Porter, Christian, Piero Sarti +5
Computer Science · Mathematics · #Analytic Number Theory Research #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2311.16870

openalex publication_date 2023/11/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we continue the study of unit reducible fields as introduced in \citeLPL23 for the special case of cyclotomic fields. Specifically, we deduce that the cyclotomic fields of conductors 2,3,5,7,8,9,12,15 are all unit reducible, and show that any cyclotomic field of conductor N is not unit reducible if 2⁴, 3³, 5², 7², 11² or any prime p ≥ 13 divide N, meaning the unit reducible cyclotomic fields are finite in number. Finally, if a is a totally positive element of a cyclotomic field, we show that for all equivalent a^′, the discrepancy between \trace_K/ℚ(a^′) and the shortest nonzero element of the quadratic form \trace_K/ℚ(axx^*) where x is taken from the ring of integers tends to infinity as the conductor N goes to infinity.

Related

• Real cyclotomic fields of prime conductor and their class numbers
• Answer to a Question of Hung and Tiep on Conductors of Cyclotomic Integers
• On the Applications of Cyclotomic Fields in Introductory Number Theory
• A Fast Multiplication Algorithm and RLWE-PLWE Equivalence for the Maximal Real Subfield of the 2^r p^s-th Cyclotomic Field
• A generalised Kummer's Conjecture