Can $\mathbb{Q×Q}$ be embedded in $\mathbb{R}$ as group? I think ans is NO : if possible let that is true hence there is a monomorphism from $H= \mathbb{Q×Q}$ to $\mathbb{R}$.  as $\mathbb{R} $ has only subgroups which is cyclic or dense and $H$ is not cyclic hence dense but it's proper subgroup 
 $\mathbb{Z×Z}$ is neither cyclic nor  dense in $\mathbb{R}$ hence contradiction.  Hence the claim. 
Is my proof correct??
Thanks. 

The map $(a,b)\mapsto a+b\sqrt2$ is an injection $\mathbb{Q×Q} \to \mathbb{R}$ because $\sqrt2$ is irrational.
$\sqrt2$ is not special here; any irrational number works, for instance $\pi$.
