Final theorem of arithmetic

Question

I have been trying to prove the "Final theorem of arithmetic" that says $\mathbb{R}$ and $\mathbb{C}$ are the only conmutative fields, $\mathbb{R}$-vectorial spaces with finite dimension. It´s easy to prove that $\mathbb{R}$ and $\mathbb{C}$ meet the conditions, but I dont know how can I prove that they are the only ones, any idea?

Answer 1

Hint: If $[K:L]$ is finite, then $K$ is an algebraic extension of $L$. $\mathbb{C}$ is algebraically-closed...