We present a new deterministic algorithm for the problem of constructing
k
k
th power nonresidues in finite fields
F
p
n
\mathbf {F}_{p^n}
, where
p
p
is prime and
k
k
is a prime divisor of
p
n
−
1
p^n-1
. We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed
n
n
and
p
→
∞
p \rightarrow \infty
, our algorithm runs in polynomial time. Unlike other deterministic algorithms for this problem, this polynomial-time bound holds even if
k
k
is exponentially large. More generally, assuming the ERH, in time
(
n
log
p
)
O
(
n
)
(n \log p)^{O(n)}
we can construct a set of elements that generates the multiplicative group
F
p
n
∗
\mathbf {F}_{p^n}^*
. An extended abstract of this paper appeared in Proc. 23rd Ann. ACM Symp. on Theory of Computing, 1991.