It is well known that the unit group of a finite field is cyclic. What can we say about the generators?
Specifically I am interested in the following question: Suppose we fix a positive integer $a$, how many primes $p$ are there such that $a$ generates the unit group $(\mathbb Z/p\mathbb Z)^\times$? Are there infinitely many?
I think this is an open problem.
There is a famous conjecture due to Artin giving the density of the set of those primes $p$ such that a non-square natural number $a$ is a primitive root modulo $p$. Positive density, of course, implies that $a$ is a primitive root modulo infinitely many primes $p$.
Note that if $a=b^\ell$ for some integer $b$ and prime $\ell$, then $a$ is automatically not a primitive root modulo a prime $p$ such that $p\equiv1\pmod\ell$. This lowers the chances for such an $a$ to be a primitive root, and the conjectured density is adjusted accordingly. An extreme case is when $a$ is a perfect square, for then this obstruction applies to all primes, and therefore perfect squares are not interesting for the purposes of this question.
There is an analogue of Artin's conjecture for the polynomial rings $R=\Bbb{F}_p[x]$. In that case the question is whether the coset of a given non-square polynomial $f(x)$ is a primitive element of the finite field $R/\langle p(x)\rangle$ for infinitely many irreducible monic polynomials $p(x)\in R$. That version (together with a density formula) was proven by Bilharz.
