Difference between Galois extension and separable extension

Question

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.

Answer 1

Notice that the field $\mathbb Q(\sqrt[3]{2})$ is contained in $\mathbb R$, but the other two roots of the irreducible polynomial $x^3-2$ are complex. Thus, the field extension contains only one of the three roots of an irreducible polynomial, thus is not normal.

Answer 2

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

But then....