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In [[mathematics]], '''Lüroth's theorem''', named after [[Jacob Lüroth]], is a result of [[field theory (mathematics)|field theory]], which is related to [[rational varieties]]. It asserts that every field extension of a field <math>K</math>, which is also a subfield of <math>K(X)</math>, is simple. 

==Statement of the theorem==
Let <math>K</math> be a field and <math>M</math> be an intermediate field between <math>K</math> and <math>K(X)</math>, for some indeterminate X. Then there exists a rational function <math>f(X)\in K(X)</math> such that <math>M=K(f(X))</math>. In other words, every
intermediate extension between <math>K</math> and <math>K(X)</math> is simple.  

The proof of Lüroth's theorem can be derived easily from the theory of rational curves, using the geometrical notion of genus. 
Even though Lüroth's theorem is generally thought to be non elementary, 
several short proofs, using only the basics of field theory, have been discovered for long.
Virtually all these simple proofs use Gauss's lemma on primitive polynomials as a main step (see e.g. <ref>{{cite journal| first=Michael|last=Bensimhoun|url = https://commons.wikimedia.org/wiki/File%3AAnother_elementary_proof_of_Luroth's_theorem-06.2004.pdf|format=PDF
| title = Another elementary proof of Lüroth's theorem|date=May 2004|}}</ref>).

==See also==
*[[Rational variety]]
*[[Rational curve]]
*[[Rational surface]]
*[[Birational geometry]]

==References==
{{Reflist}}
*{{citation|chapter=Lueroth (or Lüroth), Jakob|url=http://www.encyclopedia.com/doc/1G2-2830902699.html|first=Werner |last=Burau|title=Complete Dictionary of Scientific Biography|year=2008}}

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[[Category:Algebraic varieties]]
[[Category:Birational geometry]]
[[Category:Field theory]]