Minimal polynomial of a finite field Let $p,n\in\mathbb{N}$ with $p$ prime and $q=p^n$. Let $\mathbb{F}_q$ be the finite field with $q$ elements (unique up to $\cong$), i.e. the $q$-th Galois field. According to this, the extension $\mathbb{Z}_p\leq\mathbb{F}_q$ is simple, so we must have $$\mathbb{F}_q\cong\mathbb{Z}_p[x]/(f)$$
for some irreducible polynomial $f\in\mathbb{Z}_p[x]$. What is the formula for this $f$?

$f$ need not be unique. Take $\mathbf{Z}_5$ with $f=x^2-2,g=x^2-3$. Both give fields with $25$ elements.
