Upper bounds for cyclotomic numbers

• Main Authors: Duc, Tai Do, Leung, Ka Hin, Schmidt, Bernhard
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2020
• Subjects: Science::Mathematics; Finite Fields; Cylotomic Fields
• Online Access: https://hdl.handle.net/10356/145013
• Institution: Nanyang Technological University

Let q be a power of a prime p, let k be a nontrivial divisor of q−1 and write e=(q−1)/k. We study upper bounds for cyclotomic numbers (a,b) of order e over the finite field Fq. A general result of our study is that (a,b)≤3 for all a,b∈Z if p>(14−−√)k/ordk(p). More conclusive results will be obtained through separate investigation of the five types of cyclotomic numbers: (0,0),(0,a),(a,0),(a,a) and (a,b), where a≠b and a,b∈{1,...,e−1}. The main idea we use is to transform equations over Fq into equations over the field of complex numbers on which we have more information. A major tool for the improvements we obtain over known results is new upper bounds on the norm of cyclotomic integers.