Prove that the numbers of the form $a+b\sqrt{2}$, where $a$ and $b$ are rational numbers, form a subfield of $\mathbb{C}$.

Question

I'm having trouble proving that a multiplicative inverse exists in the following problem:

Prove that the numbers of the form $a+b\sqrt{2}$, where $a$ and $b$ are rational numbers, form a subfield of $\mathbb{C}$.

Answer 1

Hint: What happens if you multiply $(a + b \sqrt{2})$ by $(a - b \sqrt{2})$?