Let
GF
(
p
n
)
{\text {GF}}({p^n})
be the finite field with
p
n
{p^n}
elements, where p is prime. We consider the problem of how to deterministically generate in polynomial time a subset of
GF
(
p
n
)
{\text {GF}}({p^n})
that contains a primitive root, i.e., an element that generates the multiplicative group of nonzero elements in
GF
(
p
n
)
{\text {GF}}({p^n})
. 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
)
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
n = 2
. Third, we give a quantitative improvement of a theorem of Wang on the least primitive root for
GF
(
p
)
{\text {GF}}(p)
, assuming the ERH.