The quotient group $(\mathbb{R}\times \mathbb{R},+)/\{(a+b\sqrt{2},a-b\sqrt{2}):a,b\in\mathbb{Z}\}$

Question

As said in the title, I'm trying to find a representation of the quotient group $(\mathbb{R}\times \mathbb{R},+)/ \{ (a+b\sqrt{2},a-b\sqrt{2}):a,b\in\mathbb{Z} \}$ by finding a homomorphism $f$ on $\mathbb{R}\times \mathbb{R}$ with kernel equal to $\{ (a+b\sqrt{2},a-b\sqrt{2}):a,b\in\mathbb{Z} \}$, using the fundamental theorem on homomorphisms. For example, the kernels of the maps $f(x,y)=\exp(2\pi i xy)$ or $f(x,y)=\exp(2\pi i(x+y+\tfrac{x-y}{\sqrt{2}}))$ both mapping to $\{z\in\mathbb{C}:|z|=1\}$, include the subgroup $\{ (a+b\sqrt{2},a-b\sqrt{2}):a,b\in\mathbb{Z} \}$ but are not equal to it. How can I find such a homomorphism?