Let $E/F$ a field extension. If $x\in E$ is separable over $F$ is $E/F$ separable? Let $E/F$ a field extension. If $x\in E$ is separable over $F$ is $E/F$ separable ?
I would say yes since the fact that $x$ separable over $F$ implies $E(x)/F$ separable, an since $E=E(x)$ then $E/F$ would be separable. But I'm not sure if this argument is valid.

Consider $E=\mathbb{Q}(2^{1/6})$, which is clearly not separable, since it's a real field and the minimal polynomial $x^6-2$ has non-real roots. However, $\sqrt{2}\in E$ and is separable, since all the roots of $x^2-2$ are in $E$. Thus the claim is false.
What's going on here is that the selected element isn't necessarily enough to generate the whole field; it might generate a subfield. If you add the hypothesis that $F(x)=E$, then it works.
