Separable polynomial over field of characteristic 0

Let be a field of characteristic 0. Let be an irreducible polynomial. I want to prove that has no repeated roots in its splitting field. i.e. I want to show that it is separable.

It we let denote the formal derivative of .

So assume (for contradiction) that is not separable. Then it has a double root in it's splitting field, , it we call this root . Then have a common divisor of degree at least 1 in , let's call it .

I get that if this common divisor, is actually in we get a contraction (because is irreducible), but what if ?

Could anyone help?