Theorem 12.2. Assuming that:

Then

(i) If $𝔭$ ramifies in $L$ , then for every $x_{1}, \dots, x_{n} \in O_{L}$ , we have $Δ (x_{1}, \dots, x_{n}) \equiv 0 mod 𝔭$ .
(ii) If $𝔭$ is unramified in $L$ , then there exists $x_{1}, \dots, x_{n}$ such that $𝔭 ∤ (Δ (x_{1}, \dots, x_{n}))$ .