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