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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

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    It is not really clear what you are after... What exactly do you want the result of the "calculation" of the algebraic closure of that field to be? – 2017-02-15
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    I want to know how I would compute the closure of, say $\mathbb{F}_{17}$. I don't know what such a thing would look like so I can't give any more information than that – 2017-02-15
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    One does not "compute" it in any sensible way. One does things with it. Your question does not therefore make a lot of sense! What do you want to do with the algebraic closure? – 2017-02-15
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    I have an elliptic curve $E$ defined over a field $\mathbb{F}_p$ and I need to let $\overline{\mathbb{F}_p}$ be the closure of $\mathbb{F}_p$ and then choose points $P\in E\left(\overline{\mathbb{F}_p}\right)$ – 2017-02-15
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    Well, choosing points in the algebraic closure is the same thing as choosing points in finite algebraic extensions of your $F_p$ (since the algebraic closure is just the union of these) so what youy *really* wanty is to construct these finite extensions and work with them. – 2017-02-15

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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?