Previously I mentioned very briefly Granville’s The Anatomy of Integers and Permutations, which explores an analogy between prime factorizations of integers and cycle decompositions of permutations. Today’s post is a record of the observation that this analogy factors through an analogy to prime factorizations of polynomials over finite fields in the following sense.
Theorem: Let be a prime power, let be a positive integer, and consider the distribution of irreducible factors of degree in a random monic polynomial of degree over . Then, as , this distribution is asymptotically the distribution of cycles of length in a random permutation of elements.
One can even name what this random permutation ought to be: namely, it is the Frobenius map acting on the roots of a random polynomial , whose cycles of length are precisely the factors of degree of .
Combined with our previous result, we conclude that as (with tending to infinity sufficiently quickly relative to ), the distribution of irreducible factors of degree is asymptotically independent Poisson with parameters .