# Nielsen--Schreier for fields

Is it true that a subextension of a purely transcendental extension is itself purely transcendental?

In symbols, suppose we have field extensions $M/L/K$, with $M/K$ purely transcendental. Must $L/K$ be purely transcendental?

• No, that is not true. This was a conjecture of Lueroth (I do not know how to make the umlaut in comments). The first counterexamples were found by Clemens-Griffiths, with further counterexamples discovered shortly thereafter by Iskovskikh-Manin, Artin-Mumford, etc. – Jason Starr Sep 4 '17 at 7:14
• A small history point: this was not conjectured by Lüroth — Lüroth only proved the case of transcendence degree 1. Actually I do not think it was ever conjectured: the italian geometers proposed quite early a number of possible counter-examples, unfortunately with unsufficiently rigorous arguments. I do not know (and I'd be interested to know) who first called it the "Lüroth problem". – abx Sep 4 '17 at 7:24
• @abx. Thank you for the explanation. I always thought this was a conjecture. I am glad to learn the true history. – Jason Starr Sep 4 '17 at 7:28
• Ah, very interesting, thanks! Where can I find the proof of the degree one case? – Sean Eberhard Sep 4 '17 at 8:20
• @potentially dense: well, only if $K$ is algebraically closed — otherwise there are easy counter-examples. – abx Sep 4 '17 at 12:45