# Natural density of solvable quintics

A recent question asked about the topological density of solvable monic quintics with rational coefficients in the space of all monic quintics with rational coefficients. Robert Israel gave a nice proof that both solvable and unsolvable quintics are dense in $\Bbb Q^5$. A natural (non-topological) way to ask about the relative density of solvable quintics in all quintics is to ask about their natural density, as follows:

Write our quintics as $x^5+a_0x^4 + \dots + a_4$, with $a_i \in \Bbb Z$. Write $$f(N) := \frac{\text{# of solvable quintics with } |a_i| < N}{(2N+1)^5}.$$ Robert Israel's answer to the linked question gives some data that supports our intuition that yes, the solvable quintics are in fact extremely rare. Do we indeed have $\lim_{N \to \infty} f(N) = 0$? Are there known nice asymptotics for $f(N)$?

• I feel like it has been proved that 100% (in the limit) of quintics have Galois group equal to $S_5$, hence in particular are unsolvable. – Greg Martin Nov 13 '14 at 2:12
• @GregMartin I would be surprised if this wasn't the case, especially given the computational evidence for it. I'm especially interested in the asymptotics, though, as $f(N)$ seems to go to $0$ extremely quickly. – user98602 Nov 13 '14 at 2:14
• What are you looking for that isn't already in math.stackexchange.com/a/1027334/448 and the mathoverflow questions it links to mathoverflow.net/questions/58397 mathoverflow.net/questions/28453 ? – David E Speyer Dec 16 '14 at 13:36
• Nothing, @DavidSpeyer; I wasn't aware of that answer. – user98602 Dec 16 '14 at 14:53

Let $P_N$ denote the set of monic polynomials of degree $n > 0$ in $\mathbb{Z}[x]$ whose coefficients all have absolute value $< N$. S. D. Cohen gave in The distribution of Galois groups of integral polynomials (Illinois J. of Math., 23 (1979), pp. 135-152) asymptotic bounds for the ratio in the above limit, and reformulating his statement with some trivial algebra gives (at least asymptotically) that $$\frac{\#\{p \in P_N : \text{Gal}(p) \not\cong S_n\}}{N^n} \ll \frac{\log N}{\sqrt{N}};$$ note that the limit of the ratio on the right-hand side as $N \to \infty$ is $0$. This implies a fortiori for $n = 5$ that $$\lim_{N \to \infty} \frac{\#\{p \in P_N : \text{Gal}(p) \text{ is solvable}\}}{N^n} = 0,$$ since for quintic polynomials $p$, $\text{Gal}(p)$ is unsolvable iff $\text{Gal}(p) \cong A_5$ or $\text{Gal}(p) \cong S_5$.
For more see this mathoverflow.net question and this old sci.math question. Closely related questions on math.se include Is the Galois group associated to a random polynomial solvable with probability 0? and (my own) How often are Galois groups equal to $S_n$? .