Can two irreducible polynomials have a common root?

Question

I was reading proof of theorem which says

Let f(x) ∈ F[x] be an irreducible polynomial. If f'(x) $\neq$ 0 then f(x) is separable.

Proof :

Suppose r is a multiple root of f(x). Then f'(r) = 0. Since f(x) is irreducible, f(x) | f'(x). But this is a contradiction since deg f'(x) < deg f(x). Therefore f(x) is separable.

My doubt is why f(x) | f'(x) and why deg f'(x) < deg f(x) cause a trouble because we didn't assume that f(x) is the minimal polynomial of r.

Answer 1

I would prefer to go back to basics, and say that the gcd of $f$ and $f'$ has degree $\ge 1$, contradicting irreducibility.

They used instead the fact that if $f(x)$ is irreducible and $f$ and $g$ have a common root in some extension field then $f(x)$ divides $g(x)$.