Lüroth's theorem

Statement

Let $K$ be a field, and $K(t)$ be the field of univariate rational functions, i.e., the field of rational functions in one variable over $K$. Then, any subfield of $K(t)$ properly containing $K$ is of the form $K(f(t))$, where $f$ is a rational function. In particular, this subfield is again the field of univariate rational functions, with the variable now being $f(t)$.

In other words, any nontrivial sub-extension of a simple transcendental extension is again a simple transcendental extension.

Related facts

• Castelnuovo's theorem