Upper bounds on polynomials with small Galois group

Abstract

When monic integral polynomials of degree $n \geq 2$ 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, amongst such polynomials whose coefficients are bounded by $B$ in absolute value, asymptotically $(1+o(1))(2B+1)^n$ 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 $O(B^{3 - \frac{2}{p}} (\log B)^{p - 1})$ polynomials with Galois group $C_p$ and coefficients bounded by $B$.