Lüroth's theorem


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

Statement

Let be a field and be an intermediate field between and, for some indeterminate X. Then there exists a rational function such that. In other words, every
intermediate extension between and 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.
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.