Artin's conjecture on a function field

Question

Artin's conjecture states that if $a\ne -1$ and $a$ isn't a perfect square, then $a$ is a primitive root for infinitely many primes $p$. There's an analogue conjecture for function fields but what is the condition on $a$ ? Thank you !

Answer 1

The analogous conjecture over function fields is that $g \in \mathbb{F}_q[t]$ is a primitive root modulo infinitely many irreducible polynomials $P \in \mathbb{F}_q[t]$ unless $g$ is a unit or $g$ is a $k$-th power for some $k>1$ dividing $q-1$. In these cases the conjecture will trivially fail. Note that the former condition corresponds to $a \notin \{\pm1\}$ and the latter condition corresponds to requiring that $a$ is not a square in the classical setting.

Over function fields the conjecture has been proved for so-called geometric elements $g \in \mathbb{F}_q[t]$ by Herbert Bilharz in 1937 conditional on the Riemann Hypothesis over finite fields - which of course was later proved by Weil.

Chapter 10 of Michael Rosen's book 'Number Theory in Function Fields' provides an excellent exposition of the problem.