Degree of the field of rational numbers extended by a complex number

Question

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.

Answer 1

Hint:

Remember that $e^{2\pi i}=1$, now start with $\alpha =e^{\frac{2\pi i}{3}}$ and see if you can get a polynomial in $\alpha$ with rational coefficients.

Let me help you with $\alpha=\sqrt{1+i}$. $$\alpha =\sqrt{1+i}\\\alpha^2=1+i\\\alpha^2-1=i\\(\alpha^2-1)^2=-1\\\alpha^4-2\alpha^2+2=0$$ So the minimal polynomial of $\alpha=\sqrt{1+i}$ is $x^4-2x^2+2$. Can you take it from here?