Upper Bounds for Cyclotomic Numbers

Abstract

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 𝔽 q . A general result of our study is that (a,b)≤3 for all a,b∈ℤ if p>(14) k/ord k (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 𝔽 q 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.