So I want to calculate the degree of the following fields over the field of the rational numbers: $$\mathbb{Q} \left(e^{\frac{2\pi i}{3}}\right),$$
$$\mathbb {Q} \left(\sqrt{2},\sqrt{1+i}\right)$$
I know that if I extend the rational field by irrational numbers I can use the degrees of the minimal polynomials and the multiplication formula for field extensions. But with the complex numbers I don't know how to figure out the degree of the minimal polynomials.