I'm stumbled upon a statement about finite field and I hope I can get some clarifications on my confusions. The question states:
Fix a prime number p. For an integer $k \geq 1$, let $S_k(p)$ be the number of irreducible monic polynomials of degree $k$ over $\mathbb{F}_p$. For example, $S_1(p) = p$. One has $\sum_{k|r} kS_k(p) = p^r$ by the prime factorization of $x^{p^r} − x$ over $\mathbb{F}_p$. If $l$ is prime, $S_l(p) = (p^l − p)/l$.
I can't find a way to get $S_l(p) = (p^l − p)/l$ if $l$ is prime. My background knowledge about finite field limits to Chapter 15.7 in Artin, in which states that the irreducible factors of the polynomial $x^{p^r}-x$ over the prime field $F = \mathbb{F}_p$ are the irreducible polynomials in $F[x]$ whose degrees divide $r$. So I can wrap my head around $\sum_{k|r} kS_k(p) = p^r$, but how to get $S_l(p) = (p^l − p)/l$ for prime $l$.
