I want to proof the following lemma:
Given a polynomial $P \in F[X]$ the number of distinct roots is d = \deg(P) - \deg(\gcd(P,P')).
I see that if $z_1, \dots, z_n$ are the roots and $\mu_1, \dots, \mu_n$ are the multiplicities then \gcd(P,P') = (X-z_1)^{\mu_1-1} \cdots (X-z_n)^{\mu_n-1}.
Now if $P$ has exactly $\deg(P)$ roots that's fine and the lemma holds. But I do not see why it holds if there are less then $\deg(P)$ roots.
Any ideas?