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The Orders of Related Elements of a Finite Field
Authors:Cohen  Stephen D.
Affiliation:(1) Department of Mathematics, University of Glasgow, Glasgow, G12 8QW Scotland
Abstract:All finite fields 
$$mathbb{F}$$
q (q ne 2, 3, 4, 5, 7, 9, 13, 25, 121) contain a primitive element gamma for which gamma + 1/gamma is also primitive. All fields of square order 
$$mathbb{F}$$
q2 (q ne 3, 5) contain an element gamma of order q + 1 for which gamma + 1/gamma is a primitive element of the subfield 
$$mathbb{F}$$
q. These are unconditional versions of general asymptotic results.
Keywords:finite field  order  primitive element  arithmetical function  character sum
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