Hungerford defines a field, $E$ as being closed if $E=E''$ where $E'= \{ \sigma \in \mathrm{Aut}(F/K)|\sigma(u)=u \text{ for all } u\in E \} = \mathrm{Aut}(F/E)$ is a subgroup of $\mathrm{Aut}(F/K)$,
$F$ is an extension field of $K$.
I want to show that in the extension of $\mathbb{Q}$ by $\mathbb{Q}(x)$, the intermediate field $\mathbb{Q}(x^3)$ is not closed.
My Attempt. Let $u \in \mathbb{Q}(x) - \mathbb{Q}(x^3)$. Then I'd want to show that $u$ is fixed by $\sigma \in \mathrm{Aut}(\mathbb{Q}(x) / \mathbb{Q}(x^3))$. So I let $u=x^3+x$. Then $\sigma(u)=\sigma(x^3)+\sigma(x)=x^3+\sigma(x)$
Any help as usual would be appreciated.