Around , Joseph Louis Lagrange began the groundwork that unified the many different tricks that had been used up to that point to solve equations, relating them to the theory of groups of permutations , in the form of Lagrange resolvents.
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The next year, in his thesis , he wrote "After the labors of many geometers left little hope of ever arriving at the resolution of the general equation algebraically, it appears more and more likely that this resolution is impossible and contradictory. I shall set forth my investigations of this at greater length in another place. The theorem was first nearly proved by Paolo Ruffini in Abel wrote: "The first and, if I am not mistaken, the only one who, before me, has sought to prove the impossibility of the algebraic solution of general equations is the mathematician Ruffini.
But his memoir is so complicated that it is very difficult to determine the validity of his argument. It seems to me that his argument is not completely satisfying. The proof also, as it was discovered later, was incomplete. Ruffini assumed that all radicals that he was dealing with could be expressed from the roots of the polynomial using field operations alone; in modern terms, he assumed that the radicals belonged to the splitting field of the polynomial.
While Cauchy either did not notice Ruffini's assumption or felt that it was a minor one, most historians believe that the proof was not complete until Abel proved the theorem on natural irrationalities, which asserts that the assumption holds in the case of general polynomials. This was due to the fact that he had the proof printed at his own expenses and he needed to save paper and money. Proving that the general quintic and higher equations were unsolvable by radicals did not completely settle the matter, because the Abel—Ruffini theorem does not provide necessary and sufficient conditions for saying precisely which quintic and higher equations are unsolvable by radicals.
Abel was working on a complete characterization when he died in Galois was aware of the contributions of Ruffini and Abel, since he wrote "It is a common truth, today, that the general equation of degree greater than 4 cannot be solved by radicals In , Vladimir Arnold discovered a topological proof of the Abel—Ruffini theorem,    which served as a starting point for topological Galois theory.
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Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability
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