primitive roots of primes

Question

Suppse $g$ is a primitive root modulo $p$ (a prime) and suppose $m\mid{p-1} ,\ (1<m<p-1)$How many integral solutions are there of the congruence $$x^m-g\equiv{0}\ (mod\quad p)$$ So far it seems to me that since there would be infinite since the $x^m\equiv{g}(mod\quad p)$ would have an infinitely periodic solution. However, I have a strong intuition that this is terribly misguided. Thanks in advance-

Answer 1

Using Discrete Logarithm wrt primitive root $g\pmod p$,

$m\cdot $ind$_gx\equiv1\pmod{p-1}$

Now use Linear Congruence theorem, at least one solution will exist iff $(m,p-1)|1$

If $m|(p-1),(m,p-1)=m$ and if $m>1,(m,p-1)\nmid1$