Degree of a field extension over $\mathbb{C}$

Question

I am supposed to find the degree of $ \mathbb {Q} (\sqrt3) \bigcap \mathbb {Q} (i)$ over $\mathbb{C}$.

My approach: Disclaimer: not sure how to go over $\mathbb{C}$.

The minimal poly of $ \mathbb {Q} (\sqrt3) $ is $x^2 - 3 = 0$. Hence, $ \mathbb {Q} (\sqrt3) $ is of degree 2. The minimal poly of $ \mathbb {Q} (i) $ is $x^2 + 1 = 0$. So, $ \mathbb {Q} (i) $ is of degree 2. Thus, because of multiplicativity of field extensions $[ℂ:ℚ(√3)∩ℚ()] = 4$.

P.S. I found Degree of C(t) as a field extension over C, but I think it is irrelevant.

Answer 1

Since $\newcommand{\Q}{\mathbb Q} \Q \subseteq \Q(\sqrt3) \cap \Q(i) \subseteq \Bbb R \cap \Q(i) = \Q$, we have $$ \Q(\sqrt3) \cap \Q(i) = \Q. $$ Then $$ [\Bbb C : \Q(\sqrt3) \cap \Q(i)] = [\Bbb C : \Q] = \infty. $$