Suppose $E = \mathbb{Q}(2^{1/4}, i)$ and that we have an automorphism of $E$, say $\theta: E\rightarrow E$ that fixes $\mathbb{Q}$ such that $\theta(2^{1/4}) = i 2^{1/4}$. I am trying to show that $\theta(i) = i$.
I can only show that $\theta(i) = \pm i$, since $\theta(i)^2 = \theta(i^2) = \theta(-1) = -1$ so $\theta(i) = \pm i$.
However, is there anyway to show that $\theta(i) = i$?