I'm having some trouble understanding the definition of a Galois group given in class. It says that a field extension $E/F$ is a Galois extension if its group of automorphisms is such that $Fix(Aut(E/F))=F$, in other words they fix exactly the ground field.
But consider the extension $\Bbb Q( \sqrt[3]{2})/\Bbb Q$. Clearly, this is not a Galois extension, as it's not normal, but all of its automorphism still fix the ground field?
What am I misunderstanding ?