Showing $(\mathbb{Q},+)$ is not isomorphic to $(\mathbb{R},+)$ To show: $(\mathbb{Q},+)$ is not isomorphic to $(\mathbb{R},+)$
Now, the equation $x^{2} =3$ has a solution in $\mathbb{R}$, but not in $\mathbb{Q}$.  Hence they are not isomorphic to each other.  Is that right, or do I need to prove something else?
Thanks.

Recall that an isomorphism is a bijective homormophism between these two groups. Since bijections preserve the cardinality of a set, and $\Bbb{Q}$ and $\Bbb{R}$ do not have the same cardinality, there is no bijection from one to the other.
