# Separable polynomial over field of characteristic 0

Let $K$ be a field of characteristic 0. Let $f \in K[X]$ be an irreducible polynomial. I want to prove that $f$ has no repeated roots in its splitting field. i.e. I want to show that it is separable.
It we let $f'$ denote the formal derivative of $f$.
So assume (for contradiction) that $f$ is not separable. Then it has a double root in it's splitting field, $\Sigma_f$, it we call this root $\alpha \in \Sigma_f$. Then $f, f'$ have a common divisor of degree at least 1 in $\Sigma_f[X]$, let's call it $g(X)$.
I get that if this common divisor, $g(X)$ is actually in $K[X]$ we get a contraction (because $f(X)$ is irreducible), but what if $g(X) \in \Sigma_f [X] \setminus K[X]$?