Math Help - Ring Isomorphism Question

1. Ring Isomorphism Question

Hello, I am stuck with the following question:

Show that $\mathbb{R}$ is not ring-isomorphic to $\mathbb{Q}$.
Do NOT use a cardinality argument
.
Hint: suppose you had a ring isomorphism $\phi: \mathbb{R} \rightarrow \mathbb{Q}$, look at $\phi(\sqrt{2})$

$\phi(r/s)=\phi(rs^{-1})=\phi(r)\phi(s)^{-1}=\frac{\phi(r)}{\phi(s)}$
But this show we already have the rational numbers inside of the reals mapping onto the rationals, so this $\sqrt{2}$ must go into the rationals somewhere and this will contradict the injectivity of any possible isomorphism, since there are two things going to wherever phi sends $\sqrt{2}$