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 ${\mathbb{F}}_q$. A general result of our study is that $(a,b)\le 3$ for all $a,b\in \mathbb{Z}$ if $p>(\sqrt{14}{)}^{k/or{d}_k(p)}$. 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 $a\ne b$ and $a,b\in \{1,\dots ,e-1\}$. The main idea we use is to transform equations over ${\mathbb{F}}_q$ 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.