Exercise 2, § 12

Show that the number of monic polynomials of degree $n$ of $\mathbf{F}_q[X]$ which have no root in $\mathbf{F}_q$ is $q^{n-q}(q-1)^q$ for $n \geq q$ (observe that every function $\mathbf{F}_q \to \mathbf{F}_q$ is a polynomial function).