UPPER BOUNDS ON POLYNOMIALS WITH SMALL GALOIS GROUP

Abstract

When monic integral polynomials of degree n ⩾ 2 are ordered by the maximum of the absolute value of their coefficients, the Hilbert irreducibility theorem implies that asymptotically 100% are irreducible and have Galois group isomorphic to S n . In particular, among such polynomials whose coefficients are bounded by B in absolute value, asymptotically ( 1 + o ( 1 ) ) ( 2 B + 1 ) n are irreducible and have Galois group S n . When G is a proper transitive subgroup of S n , however, the asymptotic count of polynomials with Galois group G has been determined only in very few cases. Here, we show that if there are strong upper bounds on the number of degree n fields with Galois group G, then there are also strong bounds on the number of polynomials with Galois group G. For example, for any prime p, we show that there are at most O ( B 3 − 2 p ( log B ) p − 1 ) polynomials with Galois group C p and coefficients bounded by B.