Finite field as a splitting field of some irreducible polynomial

Question

In many texts that I've read regarding finite fields, it always appears to be simply stated that a finite field is a splitting field of some irreducible polynomial, without proof. What are some good sources that actually provide an explicit proof? Now on p. $3$ of the lecture notes http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/finitefields.pdf of Keith Conrad, Lemma $2.1$ states that a field of power $p^n$ (cardinality of a finite field) is a splitting field of the polynomial $x^{p^n} - x$, but I'm not sure if this is irreducible. Might this be of any help?

Answer 1

No, $x^{p^n} - x$ is not in general irreducible. What it is, though, is the product of all irreducible polynomials over $\mathbb{Z}/p$ whose degrees divide $n$. The field with $p^n$ elements will be the splitting field of any factor of $x^{p^n}-x$ that is a multiple of any of those irreducible polynomials of degree $n$.

There will be many such irreducible polynomials of degree $n$, always at least one and usually nearly $p^n/n$ of them.

In particular, even when $x^{p^n}-x$ is not itself irreducible, it is a factor of itself and does have an irreducible factor of degree n. It's the splitting field both of itself and of that factor.