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 $f$?

As far as I know, there is no "formula" to produce an irreducible polynomial of arbitrary degree over an arbitrary finite field. There are, however, algorithms, which are basically just very complicated formulas. :)
Many computer algebra systems have a function for this. For example in Pari this function is called ffinit, and the documentation says it uses "a fast variant of Adleman and Lenstra's algorithm", which I suppose refers to http://www.math.leidenuniv.nl/~hwl/PUBLICATIONS/1986a/art.pdf
