TITLE: Density of Irreducible Polynomials in $\mathbb{Z}[x]$
QUESTION [13 upvotes]: Recently I was thinking about some questions concerning $\mathbb{Z}[x]$ and realized that they might be a bit easier if I knew the relative densities of reducible polynomials.
Let $P_d$ denote the set of all elements of $\mathbb{Z}[x]$ of degree $\leq d$. My basic question is:


Question: What fraction of elements of $P_d$ factor in $\mathbb{Z}[x]$?


The fraction $f_d$ of elements of $P_d$ which factor satisfies $f_d\geq 1-\zeta(d+1)^{-1}$ (with $\zeta$ the Riemann Zeta Function) since this is the count of elements which factor as a constant times an element of $P_d$. When $d = 0,1$ one obviously has equality, but what is not clear to me is whether $f_d = 1-\zeta(d+1)^{-1}$ holds for all $d$.
I thought about the analogous problem in $\mathbb{F}_p[x]$ (where one ignores constant factors), but in this case as $d\rightarrow\infty$ one has $f_d\rightarrow 1$. If this behavior carries over to $\mathbb{Z}[x]$ then as $d$ grows large, almost every polynomial of degree $d$ factors which seems absurd; therefore looking at this question in $\mathbb{F}_p[x]$ doesn't seem to help much.
So, letting $f_d = 1-\zeta(d+1)^{-1}+\varepsilon(d)$, I am wondering if the correction term $\varepsilon$ equals zero or, if not, what is $\varepsilon(d)$ and how would one derive it?

REPLY [5 votes]: Actually, a lot more is true: a random polynomial has Galois group the full symmetric group (a result of van der Waerden) and this is true under various restrictions, and with various error terms. For a summary and some related results, see
Igor Rivin,
Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms. Duke Math. J. 142 (2008), no. 2, 353–379.  MR2401624
EDIT Concerning the result mentioned in @quid's answer: experiment shows that this is not quite sharp (at least for monic polynomials) -- the truth seems to be that the fraction of reducible polynomials is asymptotic to $1/t.$