TAOCP 4.6.2 Exercise 28
Let $N_d$ denote the number of monic irreducible polynomials of degree $d$ over $\mathbb{F}_p$.
Section 4.6.2: Factorization of Polynomials
Exercise 28. [**] [M25] The distinct-degree factorization procedure is "lucky" when there is at most one irreducible polynomial of each degree $d$; then $g(x)$ never needs to be broken into factors. What is the probability of such a lucky circumstance, when factoring a random polynomial of degree $n$, modulo $p$, for fixed $n$ as $p \to \infty$?
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Let $N_d$ denote the number of monic irreducible polynomials of degree $d$ over $\mathbb{F}p$. Then $N_d = \frac{1}{d} \sum{k \mid d} \mu(k) p^{d/k}$ by equation (26) of Section 4.6.2, where $\mu$ is the Möbius function. The distinct-degree factorization is "lucky" when each degree $d$ occurs at most once among the irreducible factors of a random polynomial of degree $n$, which requires that for all $d \le n$ the choice of irreducible factors of degree $d$ is either zero or one.
As $p \to \infty$, the total number of monic irreducible polynomials of each degree $d$ grows without bound, while the number of degree-$d$ factors in a polynomial of fixed degree $n$ remains finite. Therefore the probability that two irreducible factors of the same degree coincide tends to zero, and the probability that there is at most one factor of each degree tends to $1$. Hence for fixed $n$, the probability that the distinct-degree factorization is lucky approaches $\boxed{1}$ as $p \to \infty$.
This completes the proof.
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