On the Ruffini-Abelian theorem
James Pierpont · Bulletin of the American Mathematical Society · 1896
Gauss having rigorously established in 1799 the fundamental theorem of algebra that every equation of degree n possesses n roots real or imaginary,* it was natural to inquire more closely into the nature of these quantities.When n was less than five, it had long been known that these roots could in every case be expressed as explicit algebraic functions of the coefficients; further it was known that for every degree equations existed of more or less special nature for which this was still true.The unsuccessful attempts of the foremost mathematicians of the century which was just closing to find such expressions for the equation of degree five, when the coefficients were left indeterminate, had rendered it very doubtful if the roots of the general equation of degree greater than four possessed this property.Between the years 1799 and 1813 an Italian mathematician, Euffini,f made several attempts to establish the justice of these doubts ; but his reasoning although highly interesting and valuable is not conclusive, and the question remained open until the publication of AbePs J celebrated argument in 1826, where he proved that it was impossible to express the roots of an equation of degree greater than four, as explicit algebraic functions of the coefficients when these last were left indeterminate.Abel's demonstration, however, is not all that could be desired.In the first place, as he was ignorant of Buffini's beautiful researches, the substitution-theoretical part of his paper is unnecessarily roundabout; secondly, in the algebraical part Abel unnecessarily complicates his proof with a classification of functions according to order and degree.Here he commits an error which caused as acute a mind as Hamilton to declare that " it renders it difficult to judge of the validity of his subsequent reasoning.''In this, however, Hamilton was misled, the classification in question being entirely superfluous here, however important in other * GAUSS, Werke, vol.III.Compare also the interesting paper by Bôcher, BULLETIN, May, 1895.f Cf.BUBKHAEDT.Supplement of the Zeitschrift fur MathematiJc u.Physilc, vol.37. % Oeuvres Complètes, 2d edition, vol.I., p. 66.It is interesting to compare this with Abel's first attempt which forms the third paper of the new edition.