(In a previous question, I asked some related question but it was very disorientingly formulated. I apologize for that.)
Let $p$ be a prime and $n\geq 1$ an integer. The finite field $\mathbb{F}_{p^n}$ with $p^n$ elements may be defined (up to isomorphism) as the splitting field of the polynomial $X^{p^n}-X\in\mathbb{F}_p[X]$.
I want to find for each $p$ and $n$ an irreducible polynomial $f$ of degree $n$ in $\mathbb{F}_p[X]$ such that $\mathbb{F}_p[X]/(f)=\mathbb{F}_{p^n}$. Such a polynomial exists, for example by the primitive element theorem as $\mathbb{F}_p$ is perfect. Such a polynomial $f$ is called a primitive polynomial.
Is there a recurrence relation, like for cyclotomic polynomials, which gives such a primitive polynomial $f$ for given $p$ and $n$? (Or do I have to work for each $p$ and $n$ individually and try find some irreducible polynomial of degree $n$? If yes, what's the 'easiest' way to do it practically?)