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 [math]\displaystyle{ K }[/math] be a field and [math]\displaystyle{ M }[/math] be an intermediate field between [math]\displaystyle{ K }[/math] and [math]\displaystyle{ K(X) }[/math], for some indeterminate X. Then there exists a rational function [math]\displaystyle{ f(X)\in K(X) }[/math] such that [math]\displaystyle{ M=K(f(X)) }[/math]. In other words, every intermediate extension between [math]\displaystyle{ K }[/math] and [math]\displaystyle{ K(X) }[/math] 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]
References
- ↑ Burau, Werner (2008), "Lueroth (or Lüroth), Jakob", Complete Dictionary of Scientific Biography, http://www.encyclopedia.com/doc/1G2-2830902699.html
- ↑ Cohn, P. M. (1991), Algebraic Numbers and Algebraic Functions, Chapman Hall/CRC Mathematics Series, 4, CRC Press, p. 148, ISBN 9780412361906, https://books.google.com/books?id=KE4jqiw37PoC&pg=PA148.
- ↑ E.g. see Mines, Ray; Richman, Fred (1988), A Course in Constructive Algebra, Universitext, Springer, p. 148, ISBN 9780387966403, https://books.google.com/books?id=fMsdixbW4NoC&pg=PA148.
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