Lüroth theorem

The set of Lüroth is a result of the algebra . It was published by Jacob Lüroth in 1875.

statement

Let be a purely transcendent extension of the body of degree of transcendence 1. If an intermediate body that is different from is also purely transcendent of degree of transcendence 1. In particular, is isomorphic to . ${\displaystyle L}$ ${\displaystyle K}$ ${\displaystyle P}$ ${\displaystyle K}$ ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle L}$

General proof of this can be found in.

Other formulations

Equivalently, Lüroth's theorem can be formulated as follows: Let be a body and the body of the rational functions above , i.e. the quotient field of the polynomial ring . If an intermediate body is different from , then is for an element from . This element is always transcendent over , whereas algebraically is always over . ${\displaystyle K}$ ${\displaystyle L=K(x)}$ ${\displaystyle K}$ ${\displaystyle K[x]}$ ${\displaystyle P}$ ${\displaystyle K}$ ${\displaystyle P=K(t)}$ ${\displaystyle t}$ ${\displaystyle L}$ ${\displaystyle t}$ ${\displaystyle K}$ ${\displaystyle L}$ ${\displaystyle P}$

Another equivalent formulation in the language of algebraic geometry says that unirational curves are rational.

Lüroth problem

The question of whether Lüroth's theorem also applies to bodies with a degree of transcendence greater than one is known as the Lüroth problem. Generally this is not the case. An overview of partial results and counterexamples can be found in the book Basic Algebra II by Nathan Jacobson, cited below .

Individual evidence

1. ^ J. Lüroth: Proof of a theorem about rational curves, Math. Ann. 9: 163-165 (1875).
2. ^ "Algebraic Theory of Bodies" (1910) by Ernst Steinitz (page 302).
3. ^ N. Jacobson: Basic Algebra II (2nd. Ed.), WH Freeman, San Francisco, 1989, Sec. 8.14, pp. 520-525