 Solution

The polynomial discriminant satisfiesA polynomial has no repeated roots over the algebraic closure exactly when it is coprime to its derivative. Consequently the assumed squarefree reduction has nonzero resultant with its derivative, so $p\nmid \mathrm{disc}(f)$. On the other hand the power basis is a basis of the order in a number field $\mathbb{Z}[\alpha ]$, and the discriminant-index formula for an integral lattice givesHere the field discriminant $d_L$ is an integer. ThereforeThis is squarefree reduction excludes a prime from the power-basis index; its argument establishes the index condition before applying the Kummer-Dedekind theorem.