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

5.5. RUFFINI’s proofs of the insolubility of the quintic 89
given. 84 RUFFINI responded with another publication of a version of his proof answer-
ing to MALFATTI’S criticism; but before the discussion advanced further, MALFATTI
died.
5.5.3 RUFFINI’s final proof
In his fifth, and final, publication of his insolubility theorem 1813, RUFFINI recapitu-
lated important parts of LAGRANGE’S theory, in which he emphasized the distinction
between numerical and formal equality, before giving the refined version of his proof.
According to (Burkhardt, 1892, 155–156), the proof can be dissected into the following
parts comparable to the parts of the 1799 proof (see point 3 above):
1. If two functions y and P of the roots x1, . . . , x5 of the quintic are related by
y p − P = 0
(for any p) and P remains unaltered by the cyclic permutation (12345), there
must exist a value y1 of y which in turn changes into y2, y3, y4, and y5. Conse-
quently,
where β is a fifth root of unity.
y k = β k y1
2. If P is furthermore unaltered by the cyclic permutation (123), then y1 must
change into γy1 where γ is a third root of unity.
3. The permutation (13452) is comprised of the two cycles (12345) (123) and y must
remain unaltered. Therefore, β 5 γ 5 = 1 which in turn implies that γ = 1, demon-
strating that y cannot be altered by any of the permutations (123), (234), (345),
(451), or (512). By combining these 3-cycles the 5-cycle (12345) can be obtained,
and thus y cannot be altered by the 5-cycle, neither.
4. Consequently, it is impossible by sequential root extractions to describe func-
tions which have more than two values, and the insolubility is demonstrated.
5.5.4 Reactions to RUFFINI’s proofs
In a paper published 1845, 85 P. L. WANTZEL (1814–1848) gave a fusion argument in-
corporating the permutation theoretic arguments of RUFFINI’S final proof into the setting
of ABEL’S proof. 86
84 (Malfatti, 1804).
85 (Wantzel, 1845).
86 See also (Burkhardt, 1892, 156).