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
\begin{align*}
\phi: \mathbb{Q} \longrightarrow \mathbb{R}
\end{align*}
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
\begin{align*}
\phi(q)^2 = \phi(q^2) = x = \phi(2).
\end{align*}
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.
