A generalization of Lüroth'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.
OLLIVIER, FRANCOIS and SADIK, BRAHIM
"A simple and constructive proof to a generalization of Lüroth's theorem,"
Turkish Journal of Mathematics: Vol. 46:
4, Article 11.
Available at: https://journals.tubitak.gov.tr/math/vol46/iss4/11