Lüroth's theorem

In mathematics, Lüroth's theorem asserts that every field that lies between two other fields K and K(X) must be generated as an extension of K by a single element of K(X). This result is named after Jacob Lüroth, who proved it in 1876.

Let ${\displaystyle K}$ be a field and ${\displaystyle M}$ be an intermediate field between ${\displaystyle K}$ and ${\displaystyle K(X)}$, for some indeterminate X. Then there exists a rational function ${\displaystyle f(X)\in K(X)}$ such that ${\displaystyle M=K(f(X))}$. In other words, every intermediate extension between ${\displaystyle K}$ and ${\displaystyle K(X)}$ is a simple extension.

...
Wikipedia