Why algebraic closure of $\Bbb F\_p$ is gained by adjoining primitive $n$-th ($n$ is coprime to $p$) root of unity to $\Bbb F\_p$?

I know algebraic closure of $\Bbb F\_p$ is union of $\Bbb F\_{p^n}$ where $n$ runs $n≧1$.

But I don't know why the titled statement holds.

$\Bbb F\_{p^n}＝\Bbb F\_{p} (μ\_{p^n-1})$, where $μ\_{p^n-1}$ is $p^n$-th roof of unity, but I cannot proceed from here.

Thank you in advance.