Example 2

We have $5^2 \equiv 1 \pmod{12}$, hence the residue class of 5 (mod 12) does not generate $(\mathbf{Z}/12\mathbf{Z})^*$. So the polynomial $\Phi_{12}(X) = X^4 - X^2 + 1$ is not irreducible in $F_5[X]$; in fact we have

in $F_5[X]$.