## QQ[i] is a field

Verify that $\mathbb{Q}[i]$ is a subfield of $\mathbb{C}$.

It suffices to show that $\mathbb{Q}[i]$ is closed under subtraction and division.

To that end, suppose $\alpha = p_1+p_2i,\beta = q_1+q_2 i \in \mathbb{Q}[i]$.

$\alpha - \beta = (p_1-q_1) + (p_2-q_2)i \in \mathbb{Q}[i]$.

If $\beta \neq 0$, then $q_1^2+q_2^2 > 0$.

$\frac{\alpha}{\beta} = \frac{p_1q_1-p_2q_2}{q_1^2+q_2^2} + \frac{p_1q_2 + p_2q_1}{q_1^2+q_2^2}i \in \mathbb{Q}[i]$.

So $\mathbb{Q}[i]$ is a subfield of $\mathbb{C}$.