Searching for primitive roots in finite fields

Let GF(pⁿ) be the finite field with pⁿ elements, where p is prime. We consider the problem of how to deterministically generate in polynomial time a subset of GF(pⁿ) that contains a primitive root, i.e., an element that generates the multiplicative group of nonzero elements in GF(pⁿ). We present three results. First, we present a solution to this problem for the case where p is small, i.e., p = n^O(1). Second, we present a solution to this problem under the assumption of the Extended Riemann Hypothesis (ERH) for the case where p is large and n = 2. Third, we give a quantitative improvement of a theorem of Wang on the least primitive root for GF(p), assuming the ERH.