Is $\mathbb{C}(x)$ a finitely-generated $\mathbb{C}[x]$-algebra

Question

Is $\mathbb{C}(x)$ a finitely-generated $\mathbb{C}[x]$-algebra? Here we can replace $\mathbb{C}$ by any algebraically closed field $k$.

Answer 1

No, otherwise by the transitivity of finitely-generated algebras we will have $\mathbb{C}(x)$ a finitely-generated $\mathbb{C}$-algebra. By Hilbert's Nullstellensatz (If $R$ and $K$ are fields and $R$ is a finitely generated $K-$algebra. Then $R$ is a finite extension of $K$. If $K$ is algebraically closed, then $R=K$) we will have $\mathbb{C}(x)=\mathbb{C}$ , which is absurd. Hence, $\mathbb{C}(x)$ is not a finitely-generated $\mathbb{C}[x]$-algebra.