Supriya Ghosh (Editor)

Lüroth's theorem

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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.

Contents

Statement

Let K be a field and M be an intermediate field between K and K ( X ) , for some indeterminate X. Then there exists a rational function f ( X ) ∈ K ( X ) such that M = K ( f ( X ) ) . In other words, every intermediate extension between K 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. 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.

References

Lüroth's theorem Wikipedia
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