Abel theorem

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In 1824, Niels Henrik Abel proved the theorem:

The general algebraic equation with one unknown of degree greater than 4 is insoluble in radicals, i.e. there do not exist a formula, which expresses the roots of a general equation of degree greater than four in terms of the coefficients involving the operations of addition, subtraction, multiplication, division, raising to a natural degree and extraction of roots of natural degree. [1]

This refers to the equation

\(a_0 x^n +a_1 x^{n−1} +···+a_{n−1}x+a_n =0\)

For some reason, Alekseev numerates the coefficients beginning with the last one.

The Abel theorem above seems to have nothing to do with the Abel functions nor the Abel equations. The references about original works by Abel about the Abel functions are wanted; they should be cited in the articles superfunction and Square root of factorial. Perhaps, some refs can be found in Norway [2] or in the Proofwiki [3], but the Proofwiki denotes with the term Abel theorem another theorem, namely, the theorem about the convergence of the power series:

Let \(\displaystyle \sum_{k=0}^\infty a_k\) be a convergent series in \(\R\). Then: \(\displaystyle \lim_{x \to 1^-} \left({\sum_{k=0}^\infty a_k x^k}\right) = \sum_{k=0}^\infty a_k\)

Of course, there is interesting statistical problem about the distribution of term Abel theorem among the published mathematical results. The question is terminological, but it is important to establish the convinient system of notations and to avoid the confusions.


  1. http://www.cds.caltech.edu/~nair/abel.pdf V.B.Alekseev. Abel’s Theorem in Problems & Solutions. Translated by Sujit Nair. January 30, 2005
  2. http://www.abelprisen.no/en/abel/litteratur.html Literature on Abel.
  3. http://www.proofwiki.org/wiki/ProofWiki:Mathematicians/Niels_Henrik_Abel