What is wrong with my proof? Every extension is separable? (of course not) 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.

In characteristic $0$, indeed every irreducible polynomial is separabele, because your intuition that $p'$ is a nontrivial polynomial of lower degree is fine. However, in characteristic $\
e 0$, the derivative may be the zero polynomial and thus no contradiction arises. For example consider $X^6+aX^3+b$ in characteristic $3$: It's derivative is $6X^5+3aX^2$, i.e. $0$.