On primitive elements in finite fields of low characteristic

• We prove new character sum bounds over short intervals in a function field.
• We give applications in the context of primitive elements in finite fields.
• We show low degree polynomials have the expected number of primitive elements.

We discuss the problem of constructing a small subset of a finite field containing primitive elements of the field. Given a finite field, FqnFqn, 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.