Let $\mathbb{Q}[\sqrt{2}] = $ {$a + b \sqrt{2}$ | $a, b \in \mathbb{Q} $}. Prove that $\mathbb{Q}[\sqrt{2}]$ is a subfield of $\mathbb{C}$
Show that the set $\mathbb{Q}[\sqrt{2}] = \{a + b \sqrt{2} \mid a, b \in \mathbb{Q}\}$ is a field with the usual multiplication and addition.