Distribution of Irreducible Factors of Polynomials over $\mathbb{F}_2$ Consider monic polynomials of degree $n$ over the finite field $\mathbb{F}_2$. These polynomials can be written in the form $$f(x) = x^n+a_{n-1}x^{n-1}+\cdots + a_1x+a_0; a_i\in \lbrace 0,1 \rbrace.$$ Let $\phi(x)$ denote a random variable in this set of polynomials, with uniform distribution: $$\mathbb{P}(\phi = f)=\frac{1}{2^n}.$$ 
Factor $\phi$ into a product of irreducible factors, and let $C_i$ denote the number of irreducible factors of degree $i$.
I am interested in the limiting distribution of the process $(C_1,\ldots, C_n)$. What is the distribution of the process $(C_1,C_2,\ldots,C_n)$?
I am unsure of the distributions of the $C_i$'s due to the fact that two distinct polynomials can have the same value of $C_i$ for all $1\le i\le n$.
 A: Let $S_k$ be the space of polynomials of degree $k$. It is a $\Bbb F_2$-linear space of dimension $k$.
Let $P$ be an irreducible polynomial of degree $n$.
For any $k$ the multiplication by $P$ map $\mu_P : S_k \to S_{k+n}$ is a measure preserving bijection from polynomials of degree $k$ to polynomials of degree $k+n$ divisible by $P$.
Therefore, the probability that a random polynomial of degree $k$ is divisible by $P$ is $2^{-n}$ if $k \ge n$, the probability that it is twice divisible is $2^{-2n}$ if $k \ge 2n$, and so on.
So in the limit, the law of the number of occurences of $P$ follows an exponential distribution $P(N_P = m) = 2^{-mn}(1 - 2^{-n})$
Furthermore, if you take a finite number of irreducible polynomials $P_1,P_2, \ldots, P_m$, in the limit the numbers $N_{P_1},N_{P_2},\ldots,N_{P_m}$ are all independant from all the others (for any integer $B$, their bounded versions $\min(N_{P_i},B)$ are independant when $k$ becomes larger than something like $(B+1)\sum \deg P_i$)
In http://oeis.org/A001037 you can get the number of irreducible polynomials of each degree.
If $a_n$ is the number of irreducible polyonmials of degree $n$ then after adding all the $C_{P_i}$ of degree $n$ together, you get (in the limit)
$P(C_n = m) = \binom {m+a_n-1}{a_n-1} 2^{-mn}(1-2^{-n})^{a_n}$
