Why is $\mathbb{Q}(\sqrt[4]{2})$ $\mathbb{Q}$-isomorpic to $\mathbb{Q}(i\sqrt[4]{2})$

Question

I would suggest this is true since a isomorphism exist, namely $\phi:\mathbb{Q}(\sqrt[4]{2}) \to \mathbb{Q}(i\sqrt[4]{2}), \sqrt[4]{2} \mapsto i\sqrt[4]{2}$. But is it true that this is a homomorphism?