Find the number of integers $r$ such that the polynomial $x^{r}-a$ has a linear factor over $\mathbb{F}_{p^{n}}$

Question

If we have a finite field $\mathbb{F}_{p^{n}}$, how does one determine the number of integers $r$ in $\{0,1, \ldots, p^{n}-2 \}$ for which the equation:

$x^{r}=a$

has a solution for every $a \in \mathbb{F}_{p^{n}}$.

The problem also mentions that $p^{n}-1=q_{1}^{a_{1}}\ldots{q_{n}^{a_{n}}}$ for distince primes $q_{i}$.

My attempt to understand the problem:

I tried to understand the problem by considering the finite field $\mathbb{F}_{9}$ as the splitting field of $x^{2}+1$, whose root in $\mathbb{F}_{9}$ is regarded as $\alpha$. In this case we have $r \in \{1, \ldots, 7 \}$. I have been listing possibles $r$ and $a$'s trying to find a solution in $\mathbb{F}_{p^{n}}$, for example when I set $r=2$ and $a=\alpha$ then $x^{2}=\alpha$ when $x=\alpha+2$. I've doing this, but this process seems laborious and I haven't gotten much insight out of it yet.

Is there a obvious way to do the above mentioned problem?

Answer 1

Have you covered the following facts? Can you combine them to solve your problem?

1. The multiplicative group of $\Bbb{F}_{p^n}$ is cyclic of order $N=p^n-1$.
2. If $C_N$ is a cyclic group of order $N$, then for all $r$ the mapping $x\mapsto x^r$ is a homomorphism from $C_N$ to itself, and its kernel has size $\gcd(r,N)$.
3. A homomorphism $\phi:G\to G$ from a finite group $G$ to itself is surjective if and only if it is injective.