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 $B$.

中文翻译：

---

具有小伽罗瓦群的多项式的上界

当 $n\geq 2$ 阶的 monic 积分多项式按其系数绝对值的最大值排序时，希尔伯特不可约定理意味着渐近 100% 是不可约的，并且具有与 $S_n$ 同构的伽罗瓦群。特别地，在系数以绝对值$B$为界的多项式中，渐近地$(1+o(1))(2B+1)^n$是不可约的并且具有Galois群$S_n$。然而，当 $G$ 是 $S_n$ 的真传递子群时，只有在极少数情况下才能确定具有伽罗瓦群 $G$ 的多项式的渐近计数。在这里，我们证明如果具有伽罗瓦群 $G$ 的度数 $n$ 字段的数量有强上限，那么具有伽罗瓦群 $G$ 的多项式的数量也有强上限。例如，对于任何素数 $p$，