Why every extension of a characteristic zero field is separable? My problem is to prove that the splitting field of $f(X)=X^5+X+1$ is, in fact, a separable extension of a field $F$ with characteristic zero. 
After some researchs, I found that every extension of a field with characteristic zero is separable. 
The ideia is that, if $\alpha$ is a root of $f$ with multiplicity $>1$, so $\alpha$ is a root of $f'$ too. But why this proves that, in the case where $F$ has characteristic zero,  the extension is separable?
I'm just starting to study, so please be patient, hehehe.

An extension $K/F$ is separable iff every element of $K$ has a separable minimal polynomial over $F$. Minimal polynomials are irreducible, and in characteristic $0$ all irreducible polynomials are separable. Thus the result follows.
So why are minimal polynomials separable? A polynomial $f$ is separable iff $f$ and $f'$ have no nontrivial factors in common. And in characteristic $0$, if $f$ has degree at least $1$ and is irredicuble (since it's the minimal polynomial of some element in $K$), it has only trivial factors in common with $f'$.
In positive characteristic, however, even non-constant polynomials can have a derivative which is $0$ (say, for instance, $x^5 - 1$ in characteristic $5$), which is why there are non-separable extensions in those cases. Although for any finite field it turns out any finite extension is still separable. So the simplest examples of non-separable extensions are things like adjoining a fifth root of $T$ to the field $\Bbb F_5(T)$ (i.e. a root of the irreducible, non-separable polynomial $x^5 - T$).
