Upper bounds on polynomials with small Galois group

TL;DR

This paper establishes upper bounds on the number of polynomials with small Galois groups, linking field count bounds to polynomial count bounds, and providing explicit estimates for cyclic groups of prime order.

Contribution

It connects bounds on the number of degree n fields with Galois group G to bounds on polynomials with the same Galois group, offering explicit estimates for cyclic groups.

Findings

Bound on polynomials with Galois group C_p: O(B^{3 - 2/p} (log B)^{p - 1})

Link between field counts and polynomial counts for small Galois groups

Extension of asymptotic results to proper transitive subgroups of S_n

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 $B$.