Pairs of primitive elements in fields of even order

Let FqFq be a finite field of even   order. Two existence theorems, towards which partial results have been obtained by Wang, Cao and Feng, are now established. These state that (i) for any q⩾8q⩾8, there exists a primitive element α∈Fqα∈Fq such that α+1/αα+1/α is also primitive, and (ii) for any integer n⩾3n⩾3, in the extension of degree n   over FqFq there exists a primitive element α   with α+1/αα+1/α also primitive such that α   is a normal element over FqFq.Corresponding results for finite fields of odd order remain to be investigated.