Consider F := GF (q) where q = p^e and E := GF (q^2) where w is a primitive element of E.
Fix theta := w^(q - 2).

Starting point: can I always write 1 + theta as a power of theta?

If Gcd (q^2 - 1, q - 2) = 1, theta is a primitive element for E and the answer is yes.

So we can restrict to the case q mod 3 = 2 where all cubes of elements of E are powers of theta.

Is 1 + theta a cube in E? No, when e is odd; otherwise yes.

So in the remaining odd case we want to replace the role of 1 + theta by 1 + theta^j for some j, leading to the following conjecture.

Conjecture: Consider q = p^e where p = 2 mod 3 and e is odd. There exists j in {1, ..., q^2 - 1} such that Gcd (q^2 - 1, j) = 1 and 1 + theta^j is a cube in E.

Computational evidence in support of this conjecture is strong. While it does not hold for q = 5 and q = 8, such j exists for all relevant q in the range [11..10^8]. Always j is at most q - 1.

The context for the query is the study of presentations for the classical groups SU(3, q) \leq GL(d, q^2) where a generator has action described by theta.

Pointers towards a proof would be much appreciated.