mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Lüroth's theorem asserts that every

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

that lies between a field ''K'' and the rational function field ''K''(''X'') must be generated as an extension of ''K'' by a single element of ''K''(''X''). This result is named after

Jacob Lüroth Jacob Lüroth (18 February 1844, Mannheim, German Confederation, Germany – 14 September 1910, Munich, German Empire, Germany) was a German mathematician who proved Lüroth's theorem and introduced Lüroth quartics. His name is sometimes writte ...

, who proved it in 1876.

Statement

Let

K

be a field and

M

be an intermediate field between

K

and

K(X)

, for some indeterminate ''X''. Then there exists a

rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...

f(X)\in K(X)

such that

M=K(f(X))

. In other words, every intermediate extension between

K

and

K(X)

is a

simple extension In field theory, a simple extension is a field extension that is generated by the adjunction of a single element, called a ''primitive element''. Simple extensions are well understood and can be completely classified. The primitive element theore ...

Proofs

The proof of Lüroth's theorem can be derived easily from the theory of rational curves, using the

geometric genus In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds. Definition The geometric genus can be defined for non-singular complex projective varieties and more generally for complex ...

. This method is non-elementary, but several short proofs using only the basics of field theory have long been known, mainly using the concept of

transcendence degree In mathematics, a transcendental extension L/K is a field extension such that there exists an element in the field L that is transcendental over the field K; that is, an element that is not a root of any univariate polynomial with coefficients ...

. Many of these simple proofs use Gauss's lemma on primitive polynomials as a main step.E.g. se
this document
or .

References

{{DEFAULTSORT:Luroth's Theorem Algebraic varieties Birational geometry Field (mathematics) Theorems in algebraic geometry