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

6.9. Reception of ABEL’s work on the quintic 131
is not, in general, free from the irrationalities of the same order introduced by the
other radicals a (m)
1 , . . . of that order; and consequently the new radical, to which
this process conducts, is in general elevated to the order m + 1; a circumstance
which Abel does not appear to have remarked, and which renders it difficult to
judge of the validity of his subsequent reasoning.” 83
To HAMILTON, the mistake made by ABEL had obscured the validity of ABEL’S
subsequent reasoning, but the validity of the impossibility result, itself, was not ques-
tioned since HAMILTON had provided it with a proof not based on ABEL’S hierarchy.
Later, KÖNIGSBERGER would prove that ABEL’S hierarchy of algebraic expressions
could still be rescued (see below). By the end of the century, it was eventually realized
that the hierarchic structure imposed on algebraic expressions was actually superfluous
for the impossibility proof. 84
HAMILTON continued his scrutiny of ABEL’S proof by attacking ABEL’S character-
ization of functions of five quantities having five values under permutations:
“And because the other chief obscurity in Abel’s argument (in the opinion of
the present writer) is connected with the proof of the theorem, that a rational function
of five independent variables cannot have five values and five only, unless it
be symmetric relatively to four of its five elements; it has been thought advantageous,
in this paper, as preliminary to the discussion of the forms of functions of
five arbitrary quantities, to establish certain auxiliary theorems respecting functions
of fewer variables; which have served also to determine à priori all possible
solutions (by radicals and rational functions) of all general algebraic equations
below the fifth degree.” 85
Thus, HAMILTON pointed his finger directly at the two weak points of ABEL’S ar-
gument. For ABEL’S flawed proof of the central auxiliary theorem — that all occurring
radicals were rational functions of the roots — which he had proved by the hierar-
chic structure of algebraic expressions, HAMILTON substituted an argument descend-
ing and re-ascending the hierarchy of algebraic expressions. 86 The characterization
of functions of five variables having five values under permutations was also car-
ried out at length in an analysis which — following ABEL — reduced it to the study of
such functions when only four of the arguments were permuted. As ABEL had done,
HAMILTON completed his analysis of these functions through an extensive investigation
of particular classes. 87
HAMILTON employed a detailed style of presentation and extensive use of low
degree equations as examples; nevertheless, his exposition of ABEL’S result is not par-
ticularly clear and easy to grasp. 88 The degree of detail and a complicated notation
might also have obscured the main results to some of HAMILTON’S contemporaries.
83 (ibid., 248); small-caps changed into italic..
84 (J. Pierpont, 1896, 200).
85 (W. R. Hamilton, 1839, 248–249); small-caps changed into italic..
86 (ibid., 194–196).
87 (ibid., 237–246).
88 (Dickson, 1959, 179) calls it “a very complicated reconstruction of ABEL’S proof”.