Lüroth's theorem

In mathematics, Lüroth's theorem asserts that every field that lies between a field K and the rational function field 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.^[1]

Statement

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.

Proofs

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, mainly using the concept of transcendence degree.^[3] Many of these simple proofs use Gauss's lemma on primitive polynomials as a main step.^[4]