On multiple roots in algebraic closure of a given field

Question

I’m having trouble digesting a fact that popped up in a proof from Lang’s Algebra, page 247, proposition 6.1. Given an element $\alpha$ algebraic over $k$, and its minimum polynomial $f(X)$ over k, Lang proves that (apparently not using the fact that f is irreducible) all roots of $f$ have equal multiplicity, regardless of the field’s characteristic. This sounds funny, because it doesn’t seem to use any information other than the fact that the coefficients are all in $k$. Does this mean that any polynomial in Q will have roots of equal multiplicity in C? I’m not sure whether I’m having trouble understanding the theory or if I’m not fishing enough examples for myself. I understand the proof, it’s just that I can’t seem to give myself enough heuristic motivation for this fact in particular.