Intermediate fields of the simple extension $\mathbb{C}(x)$

Question

Let $\mathbb{C}(x)$ be the field of rational functions over $\mathbb{C}$. Of course $\mathbb{C}(x)$ is a field extension of $\mathbb{C}$. My question now is: are there any intermediate fields between $\mathbb{C}$ and $\mathbb{C}(x)$? If so, what can we say about their dimension? Is it always infinite?

Answer 1

A summary of the comments (excluding reuns result that they should post separately!) Below $K$ stands for an arbitrary intermediae field strictly in-between, $\Bbb{C}\subset K\subset\Bbb{C}(x)$.

1. Because $\Bbb{C}$ is algebraically closed, it has no algberaic extensions. Hence no finite extensions. Therefore $[K:\Bbb{C}]=\infty$.
2. On the other hand, if $u=f(x)/g(x)$ is an arbitrary element of $K\setminus\Bbb{C}$, $f,g\in\Bbb{C}[x]$, then $x$ is a zero of the polynomial $$ P(T):=f(T)-g(T)u\in K[T]. $$ Therefore $x$ is algebraic over $K$. Hence $[K(x):K]<\infty$. But, $K(x)=\Bbb{C}(x)$, so we can conclude that $[\Bbb{C}(x):K]<\infty$. Nothing more can be said, as we easily see that $[\Bbb{C}(x):\Bbb{C}(x^n)]=n$ for every positive integer $n$, so the extension degree can be arbitrarily high.
3. By Lüroth's theorem every intermediate field $K$ is actually a simple transcendental extension of $\Bbb{C}$. In other words, $K$ is $\Bbb{C}$-isomorphic to $\Bbb{C}(x)$.