Hello, I am stuck with the following question:

Show that $\displaystyle \mathbb{R}$ is not ring-isomorphic to $\displaystyle \mathbb{Q}$..

Do NOT use a cardinality argument

Hint: suppose you had a ring isomorphism$\displaystyle \phi: \mathbb{R} \rightarrow \mathbb{Q}$,look at$\displaystyle \phi(\sqrt{2})$

Thanks in advance