Examples of over $\Bbb Q(i)$ such that the Galois group is (i)$\Bbb Z_2\times \Bbb Z_2$ (ii) $D_4$

Question

I am trying to find irreducible and seperable polynomials over $\Bbb Q(i)$ such that its splitting field is Galois and isomorphic to :

(i) $\Bbb Z_2\times \Bbb Z_2$

(ii) $D_4$

I think I also need some justification to make it clear, could someone please help? Thanks so much!

Answer 1

For the first you just need the minimal polynomial of $\sqrt a+\sqrt b$ where you choose $a$ and $b$ sensibly.

With the second, you have the advantage that having fourth roots of unity you can use Kummer theory. Let $K=\Bbb Q(i)$. Let $L=\Bbb Q(\sqrt a)$ be some quadratic extension of $\Bbb Q$. If you choose some $\alpha\in L$ with the norm $N_{L/K}(\alpha)=1$ and which isn't a fourth power, then $L(\sqrt[4]{\alpha})/K$ is a $D_4$ extension.