TITLE: Difference between Galois extension and separable extension
QUESTION [5 upvotes]: a Galois extension is defined as an extension which is separable and normal.
I am looking for an example of an extension which is separable but not Galois. Apparently it is the case of$$\mathbb Q(\sqrt[3]{2})$$ but I don't really understand why. Can someone explain ? It is separable because $\sqrt[3]{2}$ is separable over $\mathbb Q$, but why is it not normal over $\mathbb Q$ ?
Thanks.

REPLY [2 votes]: $\eta : \mathbb{Q}(\sqrt[3]{2})\rightarrow \mathbb{C}$ given by $\eta (\sqrt[3]{2})=\sqrt[3]{2}. \omega $ is a $\mathbb{Q}$ homomorphism.
But then....