Pairs of primitive elements in fields of even order

Let Fq 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 8, there exists a primitive element α ∈ Fq such that α + 1/α is also primitive, and (ii) for any integer n 3, in the extension of degree n over Fq there exists a primitive element α with α+ 1/α also primitive such that α is a normal element over Fq. Corresponding results for finite fields of odd order remain to be investigated