What is the difference between $\mathbb{Q}(\alpha,i\alpha)$ and $\mathbb{Q}(\alpha,i)$? Where $\alpha = \sqrt[4]{10}$.

Question

As part of a bigger problem of finding the splitting field of $(x^4-10)(x^2-20)$ I ran into the two fields given in the title. First I am trying to focus on finding the splitting field of $x^4 - 10$. Can somebody explain the relationship between the two fields in the title?

Answer 1

They're equal. To see this it suffices to notice that each contain the generators of the other, i.e. that $\alpha,i\alpha\in \mathbb{Q}(\alpha,i)$ and $\alpha, i=\frac{i\alpha}{\alpha}\in \mathbb{Q}(\alpha,i\alpha)$.