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

6.9. Reception of ABEL’s work on the quintic 135
PIERRE LAURENT WANTZEL. In a short paper published in 1845, PIERRE LAURENT
WANTZEL refined ABEL’S proof by reversing the succession in which the radicals of
a supposed solution is studied like HAMILTON had done. 101 Although WANTZEL
deemed ABEL’S proof to be exact, he also found its presentation vague and compli-
cated. Nevertheless, WANTZEL gave no detailed reasons for his evaluation.
“Although his [ABEL’S ] proof is basically exact it is presented in a way so
complicated and so vague that it would not be generally permissible.” 102
Through a fusion of ABEL’S proof with the even vaguer and more insufficient proof
by RUFFINI, WANTZEL arrived at a clear and precise proof which he thought would
“lift all doubts concerning this important part of the theory of equations”. 103 Unfortu-
nately, he did not specify his “doubts”.
In his fusion proof, WANTZEL took over the most important of ABEL’S preliminary
arguments: the classification of algebraic expressions by orders and degrees and the
auxiliary theorem derived from it (see page 104). By studying any supposed solution
of the general n th degree equation and permutations of the roots, WANTZEL deduced
that the outermost root extraction would have to be a square root. 104 Continuing to
the radical of second highest order, he found that it had to remain unaltered by any 3-
cycle, and therefore by any 5-cycle. 105 At this point he reached a contradiction because
the supposed solution would thus only have two different values under all permuta-
tions of the five roots.
WANTZEL’S proof was published in the Nouvelles annales de mathematique and soon
became the widely accepted simplification of ABEL’S proof. It made no use of ABEL’S
classification of functions of five quantities, and may thus be seen as an indirect lo-
cal criticism of this classification. On the other hand, it builds directly upon ABEL’S
classification of algebraic expressions.
A. E. G. ANDERSSEN. In 1848, the Königlichen Friedrichs-Gymnasium in Breslau in-
vited its “protectors, sons, and friends” to be present at the annual exams. Included
with the invitation was a short essay written by one of the teachers; at the time, this
was not uncommon practice for German Gymnasien. 106 In the essay, A. E. G. ANDER-
SSEN 107 sought to illuminate the central arguments of ABEL’S impossibility proof.
Being largely a reproduction of ABEL’S argument with some elaboration of its briefest
arguments, the interesting parts of ANDERSSEN’S essay are his evaluations of ABEL’S
101 (Wantzel, 1845).
102 “Quoique sa démonstration soit exacte au fond, elle est présentée sous une forme trop compliquée
et tellement vague, qu’elle n’a pas été généralement admise.” (ibid., 57).
103 (ibid., 57).
104 (ibid., 62).
105 (ibid., 63–64). See also CAUCHY’S proof of the CAUCHY-RUFFINI theorem, section 5.6.
106 (Anderssen, 1848).
107 No further personal information concerning this Mr. ANDERSSEN has been accessible.