irreducible polynomials over finite field

Theorem.  Over a finite field $F$, there exist irreducible polynomials of any degree.

Proof.  Let $n$ be a positive integer, $p$ be the characteristic of $F$, ${𝔽}_p$ be the prime subfield, and $p^r$ be the order (http://planetmath.org/FiniteField) of the field $F$.  Since $p^r-1$ is a divisor of $p^{rn}-1$, the zeros of the polynomial $X^{p^r}-X$ form in  $G:={𝔽}_{p^{rn}}$  a subfield isomorphic to $F$.  Thus, one can regard $F$ as a subfield of $G$.  Because

$$[G:F]=\frac{[G:{𝔽}_p]}{[F:{𝔽}_p]}=\frac{rn}{r}=n,$$

the minimal polynomial of a primitive element of the field extension $G/F$ is an irreducible polynomial of degree $n$ in the ring $F[X].$