How to construct a field with exactly 125 elements

Question

How to construct a field with exactly n elements in general? Is there any method to do so? And In case no such field exists, how do you determine that?

Thanks in advance

Answer 1

First of all I think is usefull to recall the following result.

Let $\mathbb{K}$ be a field anf $f$ an irreducible polynomial of $\mathbb{K}[x]$. If $\alpha$ is a rooth of $f$ the the field $\mathbb{K}(\alpha)$ is isomorphisc to $\mathbb{K}[x]/(f)$

It is now clear that the field $\mathbb{K}(\alpha)$ is a vectorial space over $\mathbb{K}$ of dimension equal to the degree of $f$ .

Now let suppose that our field $\mathbb{K}$ is finite. We obtain:

If $f \in \mathbb{F}_p$ is an irreducible polynomial of degree $d$ and $\alpha$ is one of its roots, then the field $\mathbb{F}_q(\alpha)$ is isomorphic to $\mathbb{F}_p[x]/(f)$ ans so it has $p^d$ elements.

that give us a general construction for finite fields.