Let $E/F$ be a field extension, and suppose $a \in E$ is transcendental over $F$. I'm reading a proof that says $\dim_F F(a) \ge \dim_FF[x] = + \infty$ since the evaluation map $F[x] \to F [a]$, $p(x) \mapsto p(a)$ is one to one. However, I'm having trouble seeing why this is true.
My guess is that because $F[x]$ is isomorphic to a subspace of $F(a)$, the above claim is true. Some guidance would be appreciated.