Lüroth's theorem explained

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.

Statement

Let

be a field and

be an intermediate field between

and

K(X)

, for some indeterminate X. Then there exists a rational function

f(X)\inK(X)

such that

M=K(f(X))

. In other words, everyintermediate extension between

and

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.^[1] 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.^[2] Many of these simple proofs use Gauss's lemma on primitive polynomials as a main step.^[3]

Notes and References

.
Book: Lang, Serge . Algebra . 2002 . Springer New York . 978-1-4612-6551-1 . 3rd . Graduate Texts in Mathematics . 211 . New York, NY . 355 . Ch VIII.1 Transcendence bases . 10.1007/978-1-4613-0041-0.
E.g. see this document, or .