Example 1

The polynomial $\Phi_3(X) = X^2 + X + 1$ is irreducible in $F_q[X]$ if and only if $q \equiv 2 \pmod{3}$. Likewise $\Phi_4(X) = X^2 + 1$ is irreducible in $F_q[X]$ if and only if $q \equiv 3 \pmod{4}$ and for $\Phi_5 = X^4 + X^3 + X^2 + X + 1$, the irreducibility condition reads $q \equiv 2,\ 3 \pmod{5}$.