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.

Then there exists a rational function

In other words, every intermediate extension between

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.