Lüroth's theorem

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.

The proof of Lüroth's theorem can be derived easily from the theory of rational curves, using the geometric genus.[2] This method is non-elementary, but several short proofs using only the basics of field theory have long been known. Many of these simple proofs use Gauss's lemma on primitive polynomials as a main step.[3]