The impossibility of solving the general polynomial of degree $\ge 5$ by radicals is surely one of the most celebrated results in algebra. This result is known as the Abel-Ruffini theorem, although it's usually asserted that Ruffini's proof was incomplete.

Still, Paolo Ruffini's contributions to algebra seem to be neither widely known nor well understood. This is perhaps not so surprising since Ruffini's first attempted proof spanned 516 pages and the mathematical argument was difficult to follow.

A modern discussion of Ruffini's proof is found in Ayoub's article 'Paolo Ruffini's contributions to the quintic'. Accoring to Ayoub, Ruffini's argument was not a priori flawed, but it relies on several unproved non-trivial claims. More precisely, Ruffini fails to prove that the splitting field is one of the fields in the tower of radicals which corresponds to a solution expressed in radicals.

The revolutionary proofs of Abel and Galois following Ruffini paved way for group theory and Galois theory. Still, one could wonder what Ruffini actually proved in those 516 pages:

Did any significant new mathematical concepts, ideas or theorems arise from Ruffini's work on the quintic?

Cauchy seems to be one of the few mathematicians who found inspiration from Ruffini's work. In a letter dated 1821, he writes:

"... your memoir on the general resolution of equations is a work which has always seemed to me worthy of the attention of mathematicians and which, in my judgement, proves completely the insolvability of the general equation of degree $>4$. [...] In another memoir which I read last year to the Academy of Sciences, I cited your work and reminded the audience that your proofs establish the impossibility of solving equations algebraically ..."

In fact, according to Pesic, Cauchy's influential 1815 paper on permutations is clearly based on work of Ruffini. So are there other examples of his influence?

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    $\begingroup$ @Wadim: I believe it is Fourier. $\endgroup$ – timur Sep 13 '11 at 23:41
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    $\begingroup$ Fourier was with the Grand Prize. The judges also were Lacroix, Poisson, Legendre and Poinsot. The Cauchy connection is he had a previous version [split in two], and failed to report on it at an Academy meeting as he was indisposed (Jan 18), then talked on his own work at the next session (Jan 25). One conjecture is that in the interim he persuaded Galois to join the two manuscripts, and submit to the Grand Prize (Mar 1). persee.fr/web/revues/home/prescript/article/… This is irrelevant to the Ruffini query, for which 1830 is not "right after" 1821. $\endgroup$ – Junkie Sep 14 '11 at 7:51

Pierre L. Wantzel acknowledges the works of Niels H. Abel and Paolo Ruffini in the introduction to his complete proof of the insolubility of higher polynomial equations: In meditating on the researches of these two mathematicians, and with the aid of principles we established in an earlier paper, we have arrived at a form of proof which appears so strict as to remove all doubt on this important part of the theory of equations.

Heinrich Burkhard in Die Anfänge der Gruppentheorie holds the opinion that work of the Italien mathematician Pietro Abbati Marescotti could have been inspired by Paolo Ruffini's results. Vice versa it cannot be excluded that Abbati mentioned the first ideas of group theory to Ruffini, who subsequently expanded it.

Whether Ruffini himself or his friend Abbati had the first idea to apply group theory, partially based upon Lagrange's work on permutations, is not clear from their preserved correspondence but since Ruffini determined nearly all subgroups of the symmetric group $S_5$ his result belongs to the foundation of group theory. So his influence extends to class field theory (although modern terminology is connected with the name of Abel). This is certainly an accomplishment important enough to be called "significant new mathematical concepts or ideas".

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