I have a question on the discriminant by two different sides.
Let $\alpha \in \overline{\mathbb{Q}}$ be any algebraic number. Let $f(x) \in \mathbb{Q}[x]$ be its minimal polynomial and write it $f(x) = a \prod_{i=1}^d(x-\alpha_i)$, with $\alpha = \alpha_1$ and $d \geq 2$. Let also
$$D(f) := a^{2d-2}\prod_{i<j}(\alpha_i-\alpha_j)^2,$$
be the discriminant of $f(x)$.
Is it true that discriminant of the field $\mathbb{Q}(\alpha)$ is the same as $D(f)$? If not, how are these two discriminants related?
Nevermind, i found the solution.
The discriminant of $\mathbb{Q}(\alpha)$ is $discr(1,\alpha,\alpha^2,...,\alpha^{d-1})$ which, by a theorem, equals to
$$\frac1{a^{2d-2}}D(f).$$