Primitive generators for cyclic vector spaces over a Galois field

For a prime power q > 1 and an integer n ≥ 2 we consider the extension Fqn/Fq of Galois fields together with a cyclic Fq-vector space endomorphism τ . An element of Fqn is called a primitive τ-generator over Fq provided it generates the multiplicative group of Fqn , as well as the additive group of Fqn regarded as an Fq[x]-module with respect to τ . The notion of a primitive τ -generator generalizes the well known concept of a primitive normal basis generator; the latter is just a primitive σ-generator, where σ is the Frobenius automorphism of Fqn over Fq. The pair (q, n) as well as the corresponding extension Fqn/Fq are called extensive provided that for every cyclic Fq-vector space endomorphism τ of Fqn there exists a primitive τ -generator for Fqn over Fq. Our main result can be summarized as follows. We determine two distinguished (disjoint) sets, C with 5 pairs (q, n), and U with 19 pairs, such that no member of C is extensive, while every pair (q, n) which is not contained in the union C ∪U is an extensive one. The status whether a pair from U is extensive or not remains undecided. The proof is based on various combinatorial techniques: character theory of finite fields, a sieving method, as well as variations of a counting argument which culminate in a promising geometric argument.