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Prove that the additive groups $\mathbb{R}$ and $\mathbb{Q}$ are not isomorphic

Proof. Assume that $\mathbb{R}$ and $\mathbb{Q}$ are isomorphic. Then there exists a mapping which is bijective and is a group homomorphism. Let $x \in \mathbb{R}$ such that $\phi(2) = x$ and $q \in \mathbb{Q}$ such that $\phi(q) = \sqrt{x}$. Then Since $\phi$ is injective, we have that $q^2 = 2$. So this means that $q = \sqrt{2}$, which isn’t possible in $\mathbb{Q}$, a contradiction. Therefore, the additive groups $\mathbb{R}$ and $\mathbb{Q}$ are not isomorphic.