Lüroth's theorem
In mathematics, Lüroth's theorem asserts that every field that lies between two other fields K and 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 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.[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]