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?
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6$\begingroup$ 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. $\endgroup$– Jason StarrCommented Sep 4, 2017 at 7:14
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5$\begingroup$ 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". $\endgroup$– abxCommented Sep 4, 2017 at 7:24
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$\begingroup$ @abx. Thank you for the explanation. I always thought this was a conjecture. I am glad to learn the true history. $\endgroup$– Jason StarrCommented Sep 4, 2017 at 7:28
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$\begingroup$ Ah, very interesting, thanks! Where can I find the proof of the degree one case? $\endgroup$– Sean EberhardCommented Sep 4, 2017 at 8:20
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3$\begingroup$ @potentially dense: well, only if $K$ is algebraically closed — otherwise there are easy counter-examples. $\endgroup$– abxCommented Sep 4, 2017 at 12:45
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