Field with characteristic zero is vector space over $\mathbb{Q}$

Hint: Because $R$ is a field of characteristic $0$, every nonzero element of $\Bbb{Z}\subset R$ is invertible.

Once you have shown that $\Bbb{Q}\subset R$ all the vector space axioms are easily verified because $R$ is a field that contains $\Bbb{Q}$ as a subfield. Note that half of the axioms are already satisfied a priori because $R$ is a field. It might even be worth proving that in general:

If $R$ is a field and $S\subset R$ is a subfield then $R$ is a vector space over $S$.