Exercise 12, § 12

Let f be a polynomial of $\mathbf{Z}[X]$ of degree $n$ and let $p_1, p_2, p_3$ be three distinct prime numbers not dividing the leading coefficient off; let $f_1, f_2, f_3$ be the canonical images off in $\mathbf{F}_{p_1}[X], \mathbf{F}_{p_2}[X], \mathbf{F}_{p_3}[X]$ respectively. Suppose that $f_1$ is the product of a linear factor and an irreducible factor of degree $n - 1$, $f_2$ is the product of an irreducible factor of the second degree and irreducible factors of odd degrees, and finally $f_3$ is irreducible. Show that under these conditions f is irreducible in $\mathbf{Q}[X]$ and that the Galois group of the polynomial f over $\mathbf{Q}$ is isomorphic to the symmetric group $\mathfrak{S}_n$ (« Dedekind’s criterion » ; use Ex. 11 as well as I, p. 63, Prop. 9).