How is $[Q(\sqrt2, \sqrt3 ) : Q(\sqrt2)]=2$? $\mathbb{Q}$ is the rationals. I know that $\sqrt3 \notin \mathbb{Q}(\sqrt2)$ but so what? The answer to this question seems to be based upon that. Really don't understand what that means in finding the minimal polynomial.

If you know that $\sqrt{3}$ is not in $\mathbb{Q}(\sqrt{2})$, then you know the degree is greater than $1$. But $\sqrt{3}$ is a root of the equation $x^2-3=0$, which has coefficients in $\mathbb{Q}(\sqrt{2})$, so the degree is exactly $2$.
