Charles Hermite’s proof of Abel's theorem

Charles Hermite’s article “Considérations sur la résolution algébrique de l'équation du cinquième degré” (Reflections on the algebraic solution of the 5th degree equations) was published in 1842, in the journal Nouvelles Annales de Mathematiques. In his research, Hermite relies on the results obtained by Lagrange, noting that, in the process of solving an equation of the 5th degree, an intermediate equation of the 6th degree appears. His goal is to prove the impossibility of an algebraic solution to the last equation. His proof is based on the use of similar functions introduced by Lagrange. Of particular interest in Hermite's proof is his appeal to substitutions. He does neither use the substitution theory founded by Cauchy, nor the terminology introduced by the latter, but introduces his own. Using substitutions, he shows that it is impossible to represent a 6th degree equation by square or cubic factors. Key words: roots of equation, functions of roots, permutations
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