Pairs of primitive elements in fields of even order

Let F-q 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 alpha is an element of F-q such that alpha + 1/alpha is also primitive, and (ii) for any integer n >= 3, in the extension of degree n over F-q there exists a primitive element alpha with alpha + 1/alpha also primitive such that alpha is a normal element over F-q. Corresponding results for finite fields of odd order remain to be investigated. (C) 2014 Elsevier Inc. All rights reserved.