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Since Lagrange published his Reflections papers during the early 1770s — around 30 years before Ruffini took up and extended the subject — I was wondering if there were any results that were discovered in the meantime that may have aided Ruffini in making his group-theoretic discoveries related to his work on the quintic equation.

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    $\begingroup$ I don't know enough about this specific issue off hand to attempt an answer, and don't have time now to look through various references I have in an attempt to say something of significance, so I'll just mention that I assembled a lot of references in which this issue could be discussed in my answer to History of the theory of equations: John Colson. See also The development of Galois theory from Lagrange to Artin by B. Melvin Kiernan (1971), which unfortunately is behind a paywall. $\endgroup$ – Dave L Renfro Aug 7 '20 at 14:47
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    $\begingroup$ Lagrange had shown that an algebraic solution to the general quintic requires a resolvent equation that is a cubic or a quartic. Ruffini (eventually) showed that this was impossible "by close scrutiny of the values a polynomial can take when you permute its unknowns". (Source: John Derbyshire's Unknown Quantity.) No further explanaton is given. Derbyshire writes that Ruffini directly followed Lagrange's ideas and makes no mention of others intermediate contributions. $\endgroup$ – Nick Aug 7 '20 at 16:49
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    $\begingroup$ The MacTutor entry on Ruffini lists the four group-theoretic results that Ruffini used in his proof. The entry notes that "Ruffini used group theory in his work but he had to invent the subject for himself". Lagrange had written that a solution lay in a theory of permutations and Ruffini's work analyses (what we now call) permutation groups so this supports the view that he is not relying on any intermediate contributions. $\endgroup$ – Nick Aug 8 '20 at 19:07
  • $\begingroup$ Why the scare-quotes around 'prove' ? $\endgroup$ – Carl Witthoft Aug 10 '20 at 10:16
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    $\begingroup$ I take it that "inventing the subject" is wildly overstated, and that he simply developed those ideas relevant to the problem at hand. $\endgroup$ – Nick Aug 12 '20 at 17:30

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