Generators of the Cubic Extension of a Finite Field (pp. 189-202)
Authors: Stephen D. Cohen
Abstract: Let Fq3 be the cubic extension of the finite field Fq of prime power cardinality q. It is proved that, to every element 0 ∈ Fq3 (with Fq(0) = Fq3), there is a translate q+a, a ∈ Fq which is a primitive element (or multiplicative generator). The only exceptions relate to fields Fq with q = 3,7,9,13,37. This resolves a conjecture of Mills and McNay (2001). More generally, it is shown that, with the genuine exception of the nine fields Fq of cardinality q = 3,4,5,7,9,11,13,31,37 and the (theoretically)possible exception of any of a further 175 fields Fq (the largest of cardinality 9811), to every linear independent pair (01,02) in Fq3 there is a primitive element of the form q1 +aq2 for some a ∈ Fq. The corresponding theorem for quadratic extensions was established (with no exceptions) by the author in 1983.