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}.

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