probability of symmetric group It is known that a random polynomial with integer coefficients, which is
 irreducible over the rationals has the full symmetric group as its
 galoisgroup with a "high" probability.
I would like to have a formula which is more precise.
Concrtete : What is the probability, that a random polynomial with degree n,
             integer coefficients between -L and L and irreducible over the
             rationals, has the symmetric group S(n) as its galois group ?
 A: There is a lot of literature on this. One example is Gunter Malle, On the distribution of Galois groups, Journal of Number Theory, Volume 92, Issue 2, February 2002, Pages 315–329. The abstract goes, 
We propose a conjecture on the distribution of number fields with given Galois group and bounded norm of the discriminant. This conjecture is known to hold for abelian groups. We give some evidence relating the general case to the composition formula for discriminants, give a heuristic argument in favor of the conjecture, and present some computational data.
The paper also has references to earlier work on the problem, such as 
A.M. Baily, On the density of discriminants of quartic fields, J. Reine Angew. Math., 315 (1980), pp. 190–210; 
H. Cohen, F. Diaz y Diaz, M. Olivier,
Counting discriminants of number fields of degree up to four,
Lecture Notes in Computer Science, Springer-Verlag, New York/Berlin (2000) p. 269–283; 
H. Davenport, H. Heilbronn,
On the density of discriminants of cubic fields, II,
Proc. Roy. Soc. London, 322 (1971), pp. 405–420. 
But there's also J. Klüners, A counter example to Malle's conjecture on the asymptotics of discriminants, C. R. Acad. Sci. Paris, Ser. I 340 (2005), which says, 
In this Note we give a counter example to a conjecture of Malle which predicts the asymptotic behavior of the counting functions for field extensions with given Galois group and bounded discriminant. 
So it sounds like there's some work yet to be done. 
