Let $ \mathbb Q$ be the field of rational numbers. The set $\mathbb R$ of real numbers becomes a $\mathbb Q$-vector space under the usual addition of real numbers and scalar multiplication of rational number with a real number. Prove that $\mathbb R$ is not a finite-dimensional vector space over $\mathbb Q$.
Solution: Assume $\mathbb R$ is a finite-dimensional vector space over $\mathbb Q$ and assume that $dim_\mathbb Q \mathbb R = n$. Then there exists an isomorphism $\mathbb R \cong \mathbb Q^n $. However the cardinality of $|\mathbb R|$ is uncountable and the cardinality of $|\mathbb Q^n|$ is countable because $\mathbb Q$ is a countable set. This means that $\mathbb R $ cannot be isomorphic to $ \mathbb Q^n $ because there is no bijection. Thus, $\mathbb R$ is not finite-dimensional vector space over $\mathbb Q$.