Lüroth's Theorem

Theorem

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Let $K$ be a field.

Let $\map K X$ be the rational function field, for some indeterminate $X$.

Let $M$ be an intermediate field between $K$ and $\map K X$.

Then there exists a rational function $\map f X \in \map K X$ such that:

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$M = \map K {\map f X}$

In other words, every intermediate extension between $K$ and $\map K X$ is a simple extension.

Proof

By Gauss's Lemma (Polynomial Theory),

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Source of Name

This entry was named for Jacob Lüroth.

Source

• 2002: Serge Lang: Algebra: Ch. $\text {VIII}.1$: Transcendence bases