I couldn't find an easy to understand proof for $[ \overline{\mathbb{F\_{p}}} : \mathbb{F\_{p}} ] = \infty$, where $\overline{\mathbb{F\_{p}}}$ is the algebraic closure of the finite field $\mathbb{F\_{p}}$ with $p$ elements, p is a prime number. 
I know that $\overline{\Bbb{F\_p}}= \bigcup\_{n\in\Bbb N} \Bbb F\_{p^{n}}$.
It seems like a basic theorem, thanks in advance.