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Enumerate the irreducible polynomials with degree $d$ and integer coefficients $[-l,l]$ and check how many of them have a solvable galois-group. We can assume that the leading coefficient of the polynomials is positive.

Denote

$q(d,l) : $ number of irreducible polynomials with solvable galois-group

$r(d,l) : $ number od irreducible polynomials

I got the following values

d       l         q        r

5       1         0        104
        2        20       3720
        3       128      35068
        4       320     177944
        5       684     643404

6       1        52        292
        2       942      18894
        3      4730     250082

7       1         0        916
        2         8      96672

Everyone is invited to extend this table, but my question is :

Is there a reasonable estimate for the values $q$ and $r$ or an efficient method to determine $q$ and $r$ ? The brute force method quickly becomes infeasible.

Peter
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    One can already get the dramatic-looking bound $l^{d - \frac{1}{2}} \log l$ (in fact, this is a gross overestimate asymptotically) for the number of irreducible polynomials with Galois group not $S_n$; in particular, this is an upper bound for $q(d, l)$. See http://mathoverflow.net/questions/58397/the-galois-group-of-a-random-polynomial and http://math.stackexchange.com/questions/939103/how-often-are-galois-groups-equal-to-s-n . – Travis Willse Feb 05 '16 at 14:34
  • @Travis does log(l) mean $log_{10}(l)$ or $ln(l)$ ? – Peter Feb 05 '16 at 14:52
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    Since the limit is a gross overestimate---in fact, it is usually written as $\ll l^{d - \frac{1}{2}} \log l$---we can use either (again, asymptotically), though $\log$ typically refers to the natural logarithm in this sort of context. – Travis Willse Feb 05 '16 at 15:03

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