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

122 Chapter 6. Algebraic insolubility of the quintic
could be expressed by any algebraic formula, this formula would also provide the so-
lution to the general fifth degree equation by inserting a0 = 0 in that formula. Central
to the argument is that the supposed general solution formula for sixth degree equa-
tions not only produces a single root, but can somehow be made to produce all the
roots of the equation. This was a recurring idea in ABEL’S work on the theory of equa-
tions (see for instance theorem 10), which linked the concepts of satisfiability (a single
root could be found) and solubility (all roots could be found).
6.7 ABEL and RUFFINI
According to ABEL and his commentators, ABEL was unaware of the proofs published
by RUFFINI when he published his proofs of the impossibility result in 1824 and 1826. 43
Since questions of priority are a frequently recurring theme in the history of mathe-
matics, this independence of results is noticed by most biographers of ABEL. 44 It is my
firm conviction — based on the mathematical contents of his proof — that ABEL devel-
oped his proof independently of RUFFINI. However, the primary sources of informa-
tion on ABEL’S independence of RUFFINI are limited. The only mention of RUFFINI
made by ABEL is in his notebook entry on the theory of solubility (see chapter 8), in
the introduction to which he described RUFFINI’S proof:
“The first person, and if I am not mistaken, the only one prior to me, who has
tried to prove the impossibility of the algebraic solution of the general equations,
is the geometer Ruffini; but his memoir is so complicated that it is very difficult
to judge the validity of his reasoning. It seems to me that his reasoning is not
always satisfying. I think that the proof I gave in the first issue of this journal
[CRELLE’S Journal] leaves nothing to be desired as to rigor, but it does not have all
the simplicity of which it is susceptible. I have reached another proof based on the
same principles, but more simple, in trying to solve a more general problem.” 45
The answer derived from “trying to solve a more general problem” was never
made available in print, though. As is documented in chapter 8, such an answer was,
indeed, indirectly obtainable from ABEL’S more general research on algebraic solu-
bility which even produced an explicit example of a particular special equation which
could not be solved.
43 (N. H. Abel, 1824b; N. H. Abel, 1826a)
44 See for instance (Bjerknes, 1885, 22–23), (Bjerknes, 1930, 23), (Ore, 1954, 89–90), (Ore, 1957, 125), and
(Stubhaug, 1996, 352–353).
45 “Le premier, et, si je ne me trompe, le seul qui avant moi ait cherché à démontrer l’impossibilité
de la résolution algébrique des équations générales, est le géomètre Ruffini; mais son mémoire est
tellement compliqué qu’il est très difficile de juger de la justesse de son raisonnement. Il me paraît
que son raisonnement n’est pas toujours satisfaisant. Je crois que la démonstration que j’ai donnée
dans le premier cahier de ce journal, ne laisse rien à désirer du côté de la rigueur; mais elle n’a pas
toute la simplicité dont elle est susceptible. Je suis parvenu à une autre démonstration, fondée sur
les mêmes principes, mais plus simple, en cherchant à résoudre un problème plus général.” (N. H.
Abel, [1828] 1839, 218).