Given $f(x) = x^4-2 \in \mathbb{Q}[x]$
a) Determine all 4 roots in $\mathbb{C}$
b) Show that there are 2 roots $\alpha$, $\beta$ of $f(x)$ in $\mathbb{C}$ such that $\mathbb{Q}(\alpha)$ and $\mathbb{Q}(\beta)$ are not the same subfield in $\mathbb{C}$
a) All I did for this part was find the roots of the equation:
$x^4-2 = 0$
$x^4 = 2$
$x = \sqrt[4]{2}, -\sqrt[4]{2}$
To find all 4 roots, I believe you just need to multiply by $i$, to get the 4 roots:
$x = \sqrt[4]{2}, -\sqrt[4]{2}, i\sqrt[4]{2}, -i\sqrt[4]{2}$
b) For this part, I am not really sure how to approach it. Any guidance would be appreciated. We are given the hint to intersect with $\mathbb{R}$.