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I came across the following sentence in Stevenhagen and Lenstra's wonderful little article Chebotarëv and his density theorem:

Nikolai's interest in [polynomial] resolvents led him to study Lie groups.

I have to say I don't really see how this would happen, and I'm curious what the connection that led Cebotarev from studying resolvents to studying Lie groups was. Does anyone know? I would be interested even in an educated guess by someone who knows the two areas mathematically, but perhaps not the specific historical context.

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    $\begingroup$ Although you are interested in a mathematical, and not just an historical, answer, this may also be appropriate for HSMSE (of course with an explicit mention of the cross-post on both posts). $\endgroup$
    – LSpice
    Commented Dec 20, 2023 at 19:30
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    $\begingroup$ Thanks for pointing that out, I actually wasn't aware that HSMSE existed. I'll edit here and cross-post there to get to a wider audience. $\endgroup$
    – stillconfused
    Commented Dec 20, 2023 at 19:55

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In The Resolvent Problem (1947) Chebotarev links the number of parameters in the resolvent to a Lie group theory conjecture by Cartan:

One could hope that Cartan’s hypothesis is not true. [The hypothesis on the number of parameters of simple continuous groups, implying that an equation of the $n$-th degree has a resolvent with at least $n-3$ parameters.] Then the number of parameters in the resolvent could be reduced by more than three parameters. However, in 1938 I was able to prove the correctness of Cartan’s conjecture [1].

[1] N. Tschebotaröw. Über irreguläre Darstellungen von halbeinfachen Lieschen Gruppen [On irregular representations of semi-simple Lie groups] (1938).

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