Probabilistic Galois theory for quartic polynomials
by
Rainer Dietmann
University of Stuttgart

Let E_n(H) be the number of monic integer polynomials f of degree n and with coefficients bounded in modulus by H such that the splitting field of f has a Galois group which is a proper subgroup of the symmetric group S_n. Then as shown by Gallagher, E_n(H) << H^n-1/2 logH, so almost all integer polynomials have the full symmetric group as Galois group. For n=2 and n=3 the stronger bound E_n(H) << H^n-1+epsilon is known, where the exponent apart from the epsilon is best possible. In our talk we want to show how by an elementary argument one can obtain this bound also for degree n=4.

Date received: June 7, 2008