What is the identity element for the subgroup $H=\{a+b\sqrt{2}:a,b\in\mathbb{Q},\text{$a$ and $b$ are not both zero}\}$ of the group $\mathbb{R}^*$?

Question

For the subgroup $H=\left \{a+b\sqrt{2} : a,b\in \mathbb{Q},\text{ $a$ and $b$ are not both zero}\right \}$ of the group $\mathbb{R}^{*}$, what is its identity element? I know it must be $1$ since that is the identity element for $\mathbb{R}^{*}$. I am not sure how to make $a+b\sqrt{2}=1$. Any ideas?

My idea is that since it says "not both zero", one of them can be zero, and so $b = 0$ and $a = 1$?

Answer 1

From $a+b\sqrt2=1$, you can derive it: if $b \neq 0$, then $\sqrt2=\frac{1-a}{b} \in \mathbb{Q}$, which is absurd. Hence $b=0$ and $a=1$.