Find a primitive element for the extension $Q(3^{1/4}, i)/Q$
So, I was guessing the primitive element is $3^{1/4}+i$, and I don't have any trouble to show that $Q(3^{1/4}+i)$ is subset of $Q(3^{1/4}, i)$.
Next, we need to show $Q(3^{1/4}, i)$ is subset of $Q(3^{1/4}+i)$, which means we need to show $3^{1/4}\in{Q(3^{1/4}+i)}$ and $i\in{Q(3^{1/4}+i)}$.
I have no clue how to do it. Can any one help me on this question?
By Eisenstein you can show $(x-i)^4-3$ is minimal polynomial of $3^{1/4}+i$ $\mathbb{Q}[i]$. And so, $[\mathbb{Q}(3^{1/4}+i):\mathbb{Q}(i)]=4$. So, as $[\mathbb{Q}(i):\mathbb{Q}]=2$ we've $[\mathbb{Q}(3^{1/4}+i):\mathbb{Q}]=8$ and so you get what you want