I'm trying to prove that every extension of $\mathbb Q$ is separable.
I take an extension $E$ of $\mathbb Q$. Let $\alpha\in E$ be algebraic over $\mathbb Q$ and $p(x)$ be its minimal polynomial over $\mathbb Q$.
Suppose that the multiplicity of $\alpha$ is $m \gt 1$.
Taking the derivative $p'(x)$, we have $\alpha$ a root of $p'(x)$ which is of lower degree than $p(x)$, contradiction. Then $E$ is a separable extension of $\mathbb Q$.
I have a felling we can prove this to every field, when I'm using the fact the field is $\mathbb Q$?
Thanks a lot.