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

88 Chapter 5. Towards unsolvable equations
At this point, RUFFINI focused on the second radical to be extracted and the above
argument applied equally well to rule out the case of a fifth root. Similarly, it could
not be a square root or a fourth root since these would lead to functions having four
(2 × 2) or eight (2 × 4) values, which were proved to be non-existent. RUFFINI had
thus established that any supposed solution to the quintic equation would have to
begin with the extraction of a square root followed by the extraction of a third root.
However, as he laboriously proved by considering each case in turn, the six-valued
function obtained by these two radical extractions did not become three-valued after
the initial square root had been adjoined.
The proof which RUFFINI gave for the insolubility of the quintic was thus based on
three central parts:
1. The classification of permutations into types (table 5.1)
2. A demonstration, based on (1), that no function of the five roots of the general
quintic could have 3, 4, or 8 values under permutations of the roots.
3. A study of the two inner-most (first) radical extractions of a supposed solution
to the quintic, in which the result of (2) was used to reach a contraction.
The mere extent of the classification and the caution necessary to include all cases 82
combined with RUFFINI’S intellectual debt to LAGRANGE may serve to view RUFFINI’S
work as filling in some of the “infinite labor” described by WARING and LAGRANGE
in expressing their doubts about the solubility of higher degree equations (see section
5.4.1 above). However, RUFFINI’S investigations led to the complete reverse result:
that the solution of the quintic was impossible.
One of RUFFINI’S friends and critical readers, P. ABBATI (1768–1842), gave sev-
eral improvements of RUFFINI’S initial proof. The most important one was that he
replaced the laborious arguments based on thorough consideration of particular cases
by arguments of a more general character. 83 These more general arguments greatly
simplified RUFFINI’S proofs that no function of the five roots of the quintic could have
3, 4, or 8 different values. ABBATI was convinced of the validity of RUFFINI’S result
but wanted to simplify its proof, and RUFFINI incorporated his improvements into
subsequent proofs, from 1802 and henceforth.
Others, however, were not so convinced of the general validity of RUFFINI’S re-
sults. Mathematicians belonging to the “old generation” were somewhat stunned
by the non-constructive nature of the proofs, which they described as “vagueness”.
For instance, the mathematician G. F. MALFATTI (1731–1807) severely criticized RUF-
FINI’S result since it contradicted a general solution which he, himself, previously had
82 According to (Burkhardt, 1892, 135), RUFFINI actually missed the subgroup generated by the cycles
(12345) and (132).
83 (ibid., 140).