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

50 Chapter 4. The position and role of ABEL’s works within the discipline of algebra
as ABEL’S greatest legacy. 1
4.1 Outline of ABEL’s results and their structural
position
In the penultimate year of the eighteenth century, the Italian P. RUFFINI (1765–1822)
had published the first proof of the impossibility of solving the general quintic alge-
braically. Working within the same tradition as ABEL, RUFFINI published his investi-
gations on numerous occasions; however, his presentations were generally criticized
for lacking clarity and rigour. Not until 1826 — after ABEL had published his proof of
this result 2 — did ABEL mention RUFFINI’S proofs, and there is reason to believe that
ABEL obtained his proof independently of RUFFINI, yet from the same inspirations.
From LAGRANGE’S comprehensive study of the solution of equations 3 originated
the idea of studying the numbers of formally distinct values which a rational func-
tion of multiple quantities could take when these quantities were permuted. The idea
was cultivated and emancipated into an emerging theory of permutations by A.-L.
CAUCHY (1789–1857) who in 1815 provided the theory of permutations with its ba-
sic notation and terminology. 4 CAUCHY also established the first important theorem
within this theory when he proved a generalization of one of RUFFINI’S results to the
effect that no function of five quantities could have three or four different values under
permutations of these quantities.
Insolubility of the quintic. ABEL combined the results and terminology of CAUCHY’S
theory of permutations with his own innovative investigations of algebraic expressions
(radicals). ABEL’S proof is a representation of his approach to mathematics. Once he
had realized that the quintic might be unsolvable, he was led to study the “extent”
of the class of algebraic expressions which could serve as solutions: the “expressive
power” of algebraic solutions. Following a minimalistic definition of algebraic expres-
sions, ABEL classified these newly introduced objects in a way imposing a hierarchic
structure in the class of radicals. The classification enabled ABEL to link algebraic ex-
pressions — formed from the coefficients — which occur in any supposed solution for-
mula to rational functions of the roots of the equation. By the theory of permutations,
which ABEL had taken over from CAUCHY, he reduced such rational functions to only
a few standard forms. Considering these forms individually, ABEL demonstrated —
by reductio ad absurdum — that no algebraic solution formula for the general quintic
could exist.
1 See p. 44, above.
2 (N. H. Abel, 1824b; N. H. Abel, 1826a).
3 (Lagrange, 1770–1771).
4 (A.-L. Cauchy, 1815a).