Construct a ring isomorphism and prove it is a field.

Question

$\Bbb Q(\sqrt 3)= a + b\sqrt3,\; a, b \in \Bbb Q$

Let $S$ denote $\Bbb Q\times \Bbb Q$ with normal addition and multiplication.

Construct a ring isomorphism $\phi: \Bbb Q(\sqrt3) \to S$. Prove $S$ is a field.

I can prove that $S$ is a field but I have no idea how to even begin constructing this ring isomorphism?

Answer 1

The addition in $S$ is componentwise, $$(a+\sqrt{3} b) + (c+\sqrt{3}d) =(a+c,b+d).$$ But the multiplication is a bit twisted: $$(a+\sqrt{3} b) \cdot (c+\sqrt{3}d) =(ac+3bd,ad+bc).$$