The multiplicative groups $\mathbb{Q}^\ast$ and $\mathbb{R}^\ast$ are not isomorphic

Question

Of course, since the cardinality of $\mathbb{R}$ exceeds the cardinality of $\mathbb{Q}$, there does not exist a bijection between $\mathbb{Q}^\ast$ and $\mathbb{R}^\ast$, let alone a group isomorphism.

My question is whether it is also possible to prove that these multiplicative groups are not isomorphic without using a cardinality argument.

Answer 1

In $\mathbb{R}^\ast$ every element has a cuberoot; that is, for any $a$ in $\mathbb{R}^\ast$ there is an element $b$ such that $b^3=a$. In $\mathbb{Q}^\ast$ there are elements that do not.

Answer 2

Let $f:R^*\rightarrow Q^*$ be an isomorphism, remark that if $f(x)=-1, f(x^2)=1$, since $f$ is an isomorphism, it implies that $x^2=1$ and $x=-1$ since $f(1)=1$.

There exists $x\in R^*, f(x)=2$, if $x>0, f(\sqrt x)^2=2$ impossible. If $x<0, f(\sqrt{ -x}\sqrt{-x})=f(-x)=f(-1)f(x)=-2=f(\sqrt{-x})^2$ Impossible.

Answer 3

Suppose $\phi: \mathbb{Q}^\ast \to \mathbb{R}^\ast$ is an isomorphism. Let $a \in \mathbb{Q}^\ast$ satisfy $\phi(a)=2$. For each $n$, there is an element $b \in \mathbb{Q}^\ast$ with $\phi(b)=2^{1/n}$. Now $\phi(b^n) = \phi(b)^n = (2^{1/n})^n = 2 = \phi(a)$, so $b^n = a$. It follows that $a$ is an $n$-th power for all $n$. But then it follows that $a=1$, contradiction.