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 $F$, $\mathbb{F}_{p}$ be the prime subfield, and $p^{r}$ be the order 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  $G:=\mathbb{F}_{{p^{{rn}}}}$  a subfield isomorphic to $F$.  Thus, one can regard $F$ as a subfield of $G$.  Because

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