Is the subset $\mathbb{Q}[\sqrt2]:=\{x+\sqrt{2}y\mid x,y\in\mathbb{Q}\}$ of $\mathbb{R}$ a field?
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.