If I have a field $\mathbb{F}_p$, for $p$ prime, how would I go about calculating $\overline{\mathbb{F}_p}$, the algebraic closure of $\mathbb{F}_p$?
I have read the WolframMathWorld page which states
The field $\overline{\mathbb{F}}$ is called an algebraic closure of $\mathbb{F}$ if $\overline{\mathbb{F}}$ is algebraic over $\mathbb{F}$ and if every polynomial $f(x)\in\mathbb{F}[x]$ splits completely over $\overline{\mathbb{F}}$, so that $\overline{\mathbb{F}}$ can be said to contain all the elements that are algebraic over $\mathbb{F}$.
However I am struggling to understand how to actually apply this.
Can someone give me one (or more) concrete examples of the closure of various sets $\mathbb{F}_p$ so I can attempt to understand it a bit better please?
I am self-teaching so assume limited knowledge of complex maths and theorems
One can show that algebraic closure is unique up to isomorphism. If you happen to know about the fields $\Bbb F_{p^{n}}$ then one can simply write out the algebraic closure of $\Bbb F_p$: $$ \overline{\Bbb{F_p}}= \bigcup_{n\in\Bbb N} \Bbb F_{p^{n!}} $$ Can you show by yourself that it has the desired properties?