The following is a problem from I. Martin Isaac's Algebra. Let $E=\mathbb{Q}(\sqrt[4]{2}+i)$. I am trying to show $\mathbb{Q}(\sqrt[4]{2},i)=E$ with the following hint:
Find at least five different elements in the orbit of $i+\sqrt[4]{2}$ under $\text{Gal}(E/\mathbb{Q})$.
I have solved the problem in the usual manner i.e. by showing $\mathbb{Q}(\sqrt[4]{2}+i)\subseteq \mathbb{Q}(\sqrt[4]{2},i)$ and $\mathbb{Q}(\sqrt[4]{2},i)\subseteq \mathbb{Q}(\sqrt[4]{2}+i)$ with a few calculations.
My question is the following:
What is the theoretical framework behind Isaacs' hint?