$\mathbb{Q}(\sqrt{2}) = \{a+b\sqrt{2}: a,b\in\mathbb{Q}\}$ is a field — it is closed under addition, subtraction, multiplication, and division (every non-zero element has a multiplicative inverse: $(a+b\sqrt{2})^{-1} = \frac{a-b\sqrt{2}}{a^2-2b^2}\in\mathbb{Q}(\sqrt{2})$ when $a^2\neq 2b^2$). It is an extension field of $\mathbb{Q}$.