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.