We know that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as additive groups(as they are isomorphic as $\mathbb{Q}$ vector spaces) under the axiom of choice. But why not $(\mathbb{Q}, +)$ and $(\mathbb{Q}^2,+$) isomorphic as additive groups?
Specifically, I think for vector space isomorphism, I think since they are the smallest field to begin with(they are field of fractions of the integers), so they do not have a common basis, hence may be they are not isomorphic as vector spaces. But, what about group isomorphism? I think there is a simple counterexample. Any hints? Thanks beforehand.