Let $\displaystyle K$ be a field of characteristic 0. Let $\displaystyle f \in K[X]$ be an irreducible polynomial. I want to prove that $\displaystyle f$ has no repeated roots in its splitting field. i.e. I want to show that it is separable.

It we let $\displaystyle f'$ denote the formal derivative of $\displaystyle f$.

So assume (for contradiction) that $\displaystyle f$ is not separable. Then it has a double root in it's splitting field, $\displaystyle \Sigma_f$, it we call this root $\displaystyle \alpha \in \Sigma_f $. Then $\displaystyle f, f' $ have a common divisor of degree at least 1 in $\displaystyle \Sigma_f[X]$, let's call it $\displaystyle g(X)$.

I get that if this common divisor, $\displaystyle g(X) $ is actually in $\displaystyle K[X]$ we get a contraction (because $\displaystyle f(X)$ is irreducible), but what if $\displaystyle g(X) \in \Sigma_f [X] \setminus K[X]$?

Could anyone help?