Dimension of a vector space below I have to prove that the dimension of the vector space of real numbers over Q (rational numbers) is infinity. How can I prove? I have no idea.

Hint: Show that all $\log(p)$ for $p$ prime are linearly independent over $\mathbb{Q}$.
Apply the exponential function to the equation $\lambda_1\log(p_1)+\cdots +\lambda_n\log(p_n)=0$, and conclude that all $\lambda_i$ are zero. It follows that $\dim_{\mathbb{Q}}\mathbb{R}\ge n$ for all $n\ge 1$.
