RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

90 Chapter 5. Towards unsolvable equations
RUFFINI corresponded with CAUCHY, who in 1816 was a promising young Parisian
ingenieur. 87 CAUCHY praised RUFFINI’S research on the number of values which a
function could acquire when its arguments were permuted, a topic CAUCHY, himself,
had investigated in an treatise published the year before 1815 with due reference to
RUFFINI (see below). Following this exchange of letters CAUCHY wrote RUFFINI an-
other letter in September 1821, in which he acknowledged RUFFINI’S progress in the
important field of solubility of algebraic equations:
“I must admit that I am anxious to justify myself in your eyes on a point which
can easily be clarified. Your memoir on the general solution of equations is a work
which has always appeared to me to deserve to keep the attention of geometers.
In my opinion, it completely demonstrates the algebraic insolubility of the general
equations of degrees above the fourth. The reason that I had not lectured on it
[the insolubility ] in my course in analysis, and it must be said that these courses
are meant for students at the École Royale Polytechnique, is that I would have
deviated too much from the topics set forth in the curriculum of the École.” 88
At least by 1821, the validity of RUFFINI’S claim that the general quintic could
not be solved by radicals was propounded, not only by a somewhat obscure Italian
mathematician and the allusions of GAUSS, but also one of the most promising and
ambitious French mathematicians of the early nineteenth century. However, it should
take further publications, notably by the young ABEL, before this validity would be
accepted by the broad international community of mathematicians.
5.6 CAUCHY’ theory of permutations and a new proof of
RUFFINI’s theorem
In November of 1812, CAUCHY handed in a memoir on symmetric functions to the In-
stitut de France which was published three years later as two separate papers in the
Journal d’École Polytechnique. 89 The first of the two papers is of special interest in the
history of solubility of polynomial equations. It bears the long but precise title Mé-
moire sur le nombre des valuers qu’une fonction peut acquérir, lorsqu’on y permute de toutes
manières possibles les quantités qu’elle renferme. 90 Although CAUCHY’S issue was not
the solubility-question, his paper was to become extremely important for subsequent
research. It was primarily concerned with a more general version of RUFFINI’S result
87 (Ruffini, 1915–1954, vol. 3, 82–83).
88 “Je suis impatient, je l’avous, de me justifier à Vos yeux sur un point qui peut être facilement éclairi.
Votre mémoire sur la résolution générale des équations est un travail qui m’a toujours paru digne de
fixer l’attention des géomètres, et qui, à mon avis, démontre complètement l’insolubilité algébrique
des équations générales d’un dégré supérieur au quatrième. Si je n’en ai pas parlé dans mon cours
d’analyse, c’est que, ce cours étant destiné aux élèves d’École Royale Polytechnique, je ne devois
pas trop m’écarter des matières indiquées dans les programmes de l’école.” (ibid., vol. 3, 88–89).
89 (A.-L. Cauchy, 1815a; A.-L. Cauchy, 1815b).
90 (A.-L. Cauchy, 1815a).