TITLE: Number of irreducible polynomials of degree $r$ in $F_2[x]$
QUESTION [6 upvotes]: This is admittedly, probably an easy question for the right person here, but I cannot seem to track down an answer. The question itself may not be hard, but the answer is crucial to a math paper I am writing (and I don't know enough number theory).

True or false: There are at least exp$(r)$ irreducible polynomials in $\mathbb{F}_2[x]$ of degree $r$ or less. [I don't care about the hidden constants in the exp$(r)$-notation; could be $2^{\frac{r}{100}}$ or it could be $\frac{2^r}{r}$.]

If True a reference would be terrific. 
I am aware that there are roughly $n/\log n$ primes of size $n$ or less for $n$ large, I am trying to adapt that to a polynomial ring where the coefficients are in a characteristic 2 field.

REPLY [13 votes]: There is an old and well-known "prime number theorem" for irreducible polynomials in $\mathbb F_p[x]$ whose proof is much easier than the prime number theorem for $\mathbb Z$. You can find it in many textbooks, for example Ireland-Rosen A Classical Introduction to Number Theory Chapter 7, Section 2. 
Theorem The number of monic irreducible degree $n$ polynomials in $\mathbb F_p[x]$ is
$$ \frac{1}{n}\sum_{d\mid n} \mu(n/d)p^d. $$
It's not hard to see that the $p^n$ term in the sum dominates, so you get $O(p^n/n)$ as desired. The proof is not hard, one first shows that if we let $F_d(x)$ be the product of the monic irreducible polynomials of degree $d$, then
$$ \prod_{d\mid n} F_d(x) = x^{p^n}-x. $$