  We discuss the problem of constructing a small subset of a finite field
containing primitive elements of the field. Given a finite field,
$\mathbb{F}_{q^n}$, small $q$ and large $n$, we show that the set of all low
degree polynomials contains the expected number of primitive elements.
  The main theorem we prove is a bound for character sums over short intervals
in function fields. Our result is unconditional and slightly better than what
is known (conditionally under GRH) in the integer case and might be of
independent interest.
