A simple and constructive proof to a generalization of L\"uroth's theorem

TL;DR

This paper demonstrates that a classical proof of L{"u}roth's theorem can be extended to establish a broader generalization, showing that every transcendence degree 1 subfield of a rational function field is a simple extension.

Contribution

It provides a simple and constructive proof that extends L{"u}roth's theorem to a more general setting, enhancing understanding of subfield structures.

Findings

Classical proof applies to the generalized theorem

Every transcendence degree 1 subfield is a simple extension

Proof technique is constructive and straightforward

Abstract

A generalization of L{\"u}roth's theorem expresses that every transcendence degree 1 subfield of the rational function field is a simple extension. In this note we show that a classical proof of this theorem also holds to prove this generalization.