In mathematics, 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 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 rat ...

''K''(''X'') must be generated as an

extension Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * Ext ...

of ''K'' by a single element of ''K''(''X''). This result is named after

Jacob Lüroth Jacob Lüroth (18 February 1844, Mannheim, Germany – 14 September 1910, Munich, Germany) was a German mathematician who proved Lüroth's theorem and introduced Lüroth quartics. His name is sometimes written Lueroth, following the common pr ...

, 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 which is generated by the adjunction of a single element. Simple extensions are well understood and can be completely classified. The primitive element theorem provides a characterization of ...

Proofs

The proof of Lüroth's theorem can be derived easily from the theory of

rational curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...

s, 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 comp ...

. 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 abstract algebra, the transcendence degree of a field extension ''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of ...

. 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