The algebraic closure of a finite field has infinite dimension over that field

Question

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.

Answer 1

The algebraic closure $\overline{k}$ of any field is infinite (otherwise, $f(X) = 1 + \prod_{t\in \overline{k}}\, (X -t)$ would have no zeros); $\mathbb{F}_p$ is finite.

Answer 2

Let $K$ be a finite field. Then $K$ is a finite extension of $\Bbb{F}_p$, where $p$ is the characteristic of $K$. Therefore, the algebraic closure $\overline K$ of $K$ coincides with $\overline{\Bbb{F}_p}$.

Since you know that $\overline{\Bbb{F}_p}= \bigcup_{n} \Bbb{F}_{p^{n}}$, it follows that $\overline{\Bbb{F}_p}$ is an infinite extension of $\Bbb{F}_p$, because $[\Bbb{F}_{p^{n}}:\Bbb{F}_{p}]=n$. Therefore, $\overline{\Bbb{F}_p}$ is an infinite extension of $K$, since $K$ is a finite extension of $\Bbb{F}_p$.