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 ${\displaystyle K}$ 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. Many of these simple proofs use Gauss's lemma on primitive polynomials as a main step.[3]

References

1. Burau, Werner (2008), "Lueroth (or Lüroth), Jakob", Complete Dictionary of Scientific Biography
2. Cohn, P. M. (1991), Algebraic Numbers and Algebraic Functions, Chapman Hall/CRC Mathematics Series, 4, CRC Press, p. 148, ISBN 9780412361906.
3. E.g. see Mines, Ray; Richman, Fred (1988), A Course in Constructive Algebra, Universitext, Springer, p. 148, ISBN 9780387966403.