Let $F$ be a finite field with $q$ elements. Let $f(t) \in F[t]$ be irreducible.
(a) Prove that $f(t)$ divides $t^{2}-t$ if and only if deg $f$ divides $n$.
(b) Show that
$$
\iota^{\varsigma}-t=\prod_{\Delta \mid n} \prod_{f, w v} f_{d}(t)
$$
where the product on the inside is over all irreducible polynomials of degree $d$ with leading coefficient $1 .$
(c) Let $\psi(d)$ be the number of irreducible polynomials over $F$ of degree $d$ Show that
$$
q^{*}=\sum_{a \mid=} d \dot{\psi}(d) .
$$
(d) Let $\mu$ be the Moebius function. Prove that
$$
m \psi(n)=\sum_{a \mid \hbar} \mu(d) q^{n \cdot a} .
$$
Dividing by $n$ yields an explicit formula for the number of irreducible polynomials of degree $n$, and leading coefficient 1 over $F$.