Prove that the additive groups R and Q are not isomorphic

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.