Let $ \theta $ be an algebraic integer and let $g(t) \in \mathbb Z [ t ] $ be its minimal polynomial over $\mathbb Q$. Let $ \bar g (t) \in \mathbb F_p [ t] $ be the same polynomial with coefficients reduced mod $p$. How to prove the discriminant is unchanged, i.e. $ \overline {\text{disc} ( g (t) ) } = \text{disc} ( \bar g (t) ) $?
Of course $ \text{disc } (g (t)) \in \mathbb Z$. And is equal to $ \Pi_{ i < j } ( x_j - x_i ) $ where $x_1,...,x_n $ are roots of $g(t)$ in some splitting field. But I can't go any further, any help is appreciated.