Lüroth's theorem

In mathematics, Lüroth's theorem asserts that every field 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, who proved it in 1876.[1]

Statement

Let K {\displaystyle K} be a field and M {\displaystyle M} be an intermediate field between K {\displaystyle K} and K ( X ) {\displaystyle K(X)} , for some indeterminate X. Then there exists a rational function f ( X ) ∈ K ( X ) {\displaystyle f(X)\in K(X)} such that M = K ( f ( X ) ) {\displaystyle M=K(f(X))} . In other words, every intermediate extension between K {\displaystyle K} and K ( X ) {\displaystyle K(X)} 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, mainly using the concept of transcendence degree.[3] Many of these simple proofs use Gauss's lemma on primitive polynomials as a main step.[4]

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, vol. 4, CRC Press, p. 148, ISBN 9780412361906.
  3. ^ Lang, Serge (2002). "Ch VIII.1 Transcendence bases". Algebra. Graduate Texts in Mathematics. Vol. 211 (3rd ed.). New York, NY: Springer New York. p. 355. doi:10.1007/978-1-4613-0041-0. ISBN 978-1-4612-6551-1.
  4. ^ E.g. see this document, or Mines, Ray; Richman, Fred (1988), A Course in Constructive Algebra, Universitext, Springer, p. 148, ISBN 9780387966403.