Is the field of complex numbers the field of rational functions over a subfield?

Question

My question is the same as the title. In fact, it is another question which leads me to think about this question: For which positive integer $n$ for which there is a subfield $K$ of $\mathbb{C}$ such that $[\mathbb{C}:K]=n$. I tend to believe this is true for every $n$, and it is indeed the case if my first question is affirmative. Thank you.