Number of solutions to $x^{3^{n+1}+1} = 1$ in a field of order $3^{2n+1}$

Question

Let $F$ be a field such that $|F|=3^{2n+1}$ and $r=3^{n+1}$. I want to find the number of $x\in F$ that satisfies the equation $x^{r+1}=1$.

Answer 1

Well, since the multiplicative field of $\;\Bbb F\;$ , namely $\;\Bbb F^*:=\Bbb F-\{0\}\;$ is cyclic of order $\;3^{2n+1}-1\;$, we get that forall

$$x\in\Bbb F^*\;,\;\;x^{3^{2n+1}-1}=1\;$$

You though want some $\;x\in\Bbb F^*\;$ s.t.

$$x^{3^{n+1}+1}=1\iff \left(3^{n+1}+1\right)\mid\left(3^{2n+1}-1\right)$$

Try to take it from here.