Here is the function I am speaking about:
Recall that the field $\mathbb{Q}(\sqrt{3})$ has basis $\{1 ,\sqrt 3\}$ as a vector space over $\mathbb{Q}.$ Let $\mathbb{Z}[\sqrt{3}] \subset \mathbb{Q}(\sqrt{3})$ be the subring $\{ a_{1} + a_{2}\sqrt{3}| a_{1},a_{2} \in \mathbb{Z}\}.$
Show that the function $$\phi:\mathbb{Q}(\sqrt{3}) \rightarrow M_{2}(\mathbb{Q}), \phi (a_{1} + a_{2}\sqrt{3}) = \begin{pmatrix} a_{1} & 3a_{2} \\ a_{2} & a_{1} \end{pmatrix} $$ is an injective ring homomorphism.
My question is:
I have no problem in showing the following:
1- Showing that it is injective.
2- Showing that it preserves addition.
3- Showing that it preserves multiplication.
But I have a problem in
4- Showing that it maps the identity element of $\mathbb{Q}(\sqrt{3})$ to the identity element of $M_{2}(\mathbb{Q}).$
I know the identity element of $M_{2}(\mathbb{Q})$ which is $$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$ but it is not clear for me what is the identity element of $\mathbb{Q}(\sqrt{3}),$ could anyone clarify this for me please?
The ring $\Bbb{Q}(\sqrt{3})$ contains $\Bbb{Q}$ as a subring. Hence the identity (additive or multiplicative) of $\Bbb{Q}(\sqrt{3})$ is that of $\Bbb{Q}$.