Let $K=\mathbb{Q}(i\sqrt{2}), L=K(\sqrt[3]{6+i}), M=K(\sqrt[6]{6+i})$. Which of $K\subset L,K\subset M, L\subset M$ is a normal extension. What is their degree and find a Basis for each.
As far as I understood it for it to be a normal extension it has to be a splitting field of a polynomial in the smaller field. The problem is that I don't know how to check it and the problem arises that I don't know how elements in L and M for example look like. I would really need some help.