Exercise 23, § 11

Let $n$ and $p$ be odd and distinct prime numbers. a) Let $Q \in \mathbf{F}_p[X]$ be the image of the cyclotomic polynomial $\Phi_n$. Show that the number $i$ of irreducible factors of $Q$ is even if and only if $p$ is a square mod $n$. (Let $K$ be the splitting field of $Q$. Show that $\mathrm{Gal}(K/\mathbf{F}_p)$ may be identified with the subgroup of $(\mathbf{F}_n)^*$ generated by the image of $p$. Study the parity of the index $((\mathbf{F}_n)^*: \mathrm{Gal}(K/\mathbf{F}_p))$.) b) Show that the discriminant of $X^n - 1 \in \mathbf{F}_p[X]$ is

c) Show that $i$ is even if and only if $(-1)^{\frac{n(n-1)}{2}} n$ is a square modp (apply En. 22 to the polynomial $X^n - 1$). d) Put $\left( \frac{n}{p} \right) = 1$ if $n$ is a square modp and $\left( \phantom{\frac{n}{p}} \right) = -1$ otherwise. Deduce from c) that