Discriminant and conductor of an order of an algebraic number field
Solution 1:
Let $d$ be the discriminant of $f(X)$. Let $\mathfrak{D}_{K/\mathbb{Q}}$ be the different. Then $f'(\theta)B = \mathfrak{f}\mathfrak{D}_{K/\mathbb{Q}}$ (e.g. Neukirch, Ch. III, Th. 2.5). Taking norms of the both sides, we get $|d| = N(\mathfrak{f})|D|$, where $D$ is the discriminant of $K$.