Lüroth's theorem

In mathematics, Lüroth's theorem asserts that every field that lies between two other fields K and K(X) must be generated as an extension of K by a single element of K(X). This result is named after Jacob Lüroth, who proved it in 1876.[1]

Statement

Let be a field and be an intermediate field between and , for some indeterminate X. Then there exists a rational function such that . In other words, every intermediate extension between and is a simple extension.

Proofs

The proof of Lüroth's theorem can be derived easily from the theory of rational curves, using the geometric genus.[2] This method is non-elementary, but several short proofs using only the basics of field theory have long been known. Many of these simple proofs use Gauss's lemma on primitive polynomials as a main step.[3]

gollark: I can type VERY fast.
gollark: Isn't it dissociative?
gollark: You don't want people knowing you're my alt, so you act as if you aren't.
gollark: > how am i his altCLEARLY a disguise.
gollark: I'm using a different browser without the squareization applied.

References
  1. Burau, Werner (2008), "Lueroth (or Lüroth), Jakob", Complete Dictionary of Scientific Biography
  2. Cohn, P. M. (1991), Algebraic Numbers and Algebraic Functions, Chapman Hall/CRC Mathematics Series, 4, CRC Press, p. 148, ISBN 9780412361906.
  3. E.g. see Mines, Ray; Richman, Fred (1988), A Course in Constructive Algebra, Universitext, Springer, p. 148, ISBN 9780387966403.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.