Lüroth's theorem

Lüroth's theorem states that every field strictly between a field k and the one-variable rational function field k(x) has the form k(f) for some rational function f in k(x).

Connect

Rational function field: A field of rational functions in one or more indeterminates over a base field. The ambient field k(x) in the theorem is the one-variable case.
Function field of an algebraic curve: The field of rational functions on an algebraic curve. The theorem concerns the function field of the projective line and its intermediate subfields.
Jakob Lüroth: A German mathematician whose work included the theorem on intermediate subfields of a one-variable rational function field. The theorem bears Lüroth's name and is associated with his nineteenth-century work.
Rational curve: An algebraic curve birationally equivalent to the projective line. The theorem says the curve represented by any intermediate field is rational.
Zariski's counterexample to Lüroth's problem: A construction showing that higher-dimensional subfields of rational function fields need not be purely transcendental. It demonstrates that Lüroth's conclusion fails when the transcendence degree exceeds one.
Intermediate field: A field lying between a base field and a larger field extension. Lüroth's theorem classifies intermediate fields between k and k(x).
Projective line: The algebraic curve formed by the affine line together with one point at infinity. The field k(x) is the function field of the projective line.
Rational parametrization: A description of a curve's points by rational functions of one parameter. The theorem identifies every intermediate field with a one-parameter rational function field.
Castelnuovo's rationality criterion: A criterion for determining when a surface over an algebraically closed field is rational, using its geometric invariants. It addresses rationality in dimension two, where Lüroth's one-dimensional conclusion does not extend directly.
Stable rationality: A property of a variety that becomes rational after taking a product with projective space. Higher-dimensional rationality has weaker variants, unlike the unconditional curve conclusion here.