Is Q(4th root(2))/Q a galois extension

Question

I'm having some difficulty with the definition for galois extensions. The definition as read from my notes is $L/K$ is galois if $L^{Aut(L/K)}= K$. Where $Aut(L/K)$ is definied to be the set of all automorphisms over K, and $L^S:=${$\alpha\in L|\sigma (\alpha)=\alpha \space\space\forall \sigma\in S$}. But then I can't find an extension which isn't galois. Take this example, is $\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}$ a galois extension?

Answer 1

Call $a = \sqrt[4]{2}$. Then $\mathbb{Q}(a) / \mathbb{Q}$ is an extension of degree 4.

Consider the automorphism $\sigma \in Aut (\mathbb{Q}(a)/ \mathbb{Q} )$ which sends $a \mapsto -a$. Then $a^2 = \sqrt{2} \notin \mathbb{Q}$ is a fixed element of $\sigma$.