does there exist a transcendental extension of $\mathbb{C}$

Question

I think everyone know that $\mathbb{C}$ is an extension of $\mathbb{R}$.

Now I wondered if there exist an transcendental extension of $\mathbb{C}$

I know it's a short question but I can't find any extension, so I would say no, but can someone confirm that?

Answer 1

Yes, the field of rational functions $\mathbb C(x)$ is a transcendental extension of $\mathbb C$.

(Also, $\mathbb R(x)$ is a transcendental extension of $\mathbb R$,

whereas $\mathbb C$ is an algebraic extension of $\mathbb R$.)