LetGF(pn){\text {GF}}({p^n})be the finite field withpn{p^n}elements, wherepis prime. We consider the problem of how todeterministicallygenerate in polynomial time a subset ofGF(pn){\text {GF}}({p^n})that contains a primitive root, i.e., an element that generates the multiplicative group of nonzero elements inGF(pn){\text {GF}}({p^n}). We present three results. First, we present a solution to this problem for the case wherepis small, i.e.,p=nO(1)p = {n^{O(1)}}. Second, we present a solution to this problem under the assumption of the Extended Riemann Hypothesis (ERH) for the case wherepis large andn=2n = 2. Third, we give a quantitative improvement of a theorem of Wang on the least primitive root forGF(p){\text {GF}}(p), assuming the ERH.