Exercise 13, § 12

Let $p_1, p_2, \ldots, p_m$ be distinct odd prime numbers and $P = p_1 p_2 \ldots p_m$ ($m \geq 3$). a) Given an integer $n \geq 1$, let E be the set of polynomials in $\mathbf{Z}[X]$ of degree $\leq n$, whose coefficients lie in the interval (0, P[ of N : we thus have $\mathrm{Card}(E) = P^{n+1}$. Let k be a number such that $0 < k < 1/3n$; show that among the polynomials $f \in E$ there are at least $k^{m} P^{n+1}$ such that for every index $j$ such that $1 \leq j \leq m$, the canonical image $f_j$ of $f$ in $F_{p_j}[X]$ is of degree $n$ and irreducible (use Ex. 1, b)). Suppose moreover that $k < \frac{1}{18(n-2)}$ and $n > 2$; arguing in the same way and using Dedekind's criterion (Ex. 12) show that there exists a set of polynomials $f \in E$, of cardinal at least equal to $(1 - 3(1-k)^m)P^{n+1}$ which are of degree $n$, irreducible and whose Galois group over $\mathbf{Q}$ is isomorphic to $\mathscr{G}_s$. b) For every integer $N$ let $L_{n,N}$ be the set of polynomials of $\mathbf{Z}[X]$ of degree $\leq n$, whose coefficients belong to the interval $(-N, N)$; we thus have $\mathrm{Card}(L_{n,N}) = (2N+1)^{n+1}$. The integer $n$ being fixed, show that for every $\varepsilon$ such that $0 < \varepsilon < 1$, there exists an integer $N_0$ such that for $N \geq N_0$, there are among the polynomials $f \in L_{n,N}$ at least $(1-\varepsilon)(2N+1)^{n+1}$ which are irreducible of degree $n$, and whose Galois group over $\mathbf{Q}$ is isomorphic to $\mathfrak{S}_n$ (use a)).