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

132 Chapter 6. Algebraic insolubility of the quintic
Neither HAMILTON’S exposition of ABEL’S proof nor his more direct criticisms of JER-
RARD’S works seemed to convince JERRARD of his mistake. 89 JERRARD continued to
announce his claim in the Philosophical Magazine and in 1858 he published his Essay
on the Resolution of Equations. By that time it was left to A. CAYLEY (1821–1895) and J.
COCKLE to refute JERRARD’S claims. 90
BERNT MICHAEL HOLMBOE. The French mathematical community mainly knew
of ABEL’S work on the solubility of equations through BERNT MICHAEL HOLMBOE’S
edition of ABEL’S collected works (see above). 91 HOLMBOE’S extensive annotations
and elaborations were often supplying explicit calculations in places where ABEL had
been brief. In terms of criticism and modification of ABEL’S proof, HOLMBOE’S anno-
tations center on three topics: irreducibility, functions of five quantities, and an explicit
description of the process of inversion which ABEL had employed (see page 119).
HOLMBOE opened with a short treatment of reducible and irreducible equations,
in which he gave examples. He explicitly termed an equation irreducible when no
root of the equation could be the root of an equation of “the same form”, but of lower
degree. 92 This definition was implicit in ABEL’S paper; 93 it later took on a more explicit
and very central role in ABEL’S theory of solubility (see chapters 7 and 8).
Concerning ABEL’S investigations of functions of five quantities with five values,
HOLMBOE’S annotations are of another character giving alternative proofs of unclear
points. Remaining faithful to ABEL’S approach in the case in which µ = 2 (see page
115), HOLMBOE supplied expressions with 30 and 10 different values to rule out the
cases m = 4 and m = 5 which ABEL had left out. Thus, HOLMBOE sought to complete
ABEL’S deduction of a contradiction. But sensing the obscure nature of ABEL’S classi-
fication of functions of five quantities with five values, HOLMBOE set out to derive his
own. 94 HOLMBOE applied a general theorem, which he had proved in the Magazin for
Naturvidenskaberne:
“In the same way one can demonstrate that if u designates a given function
of n quantities which takes on m different values when one interchanges these n
quantities among themselves in all possible ways, the general form of the function
of n quantities which by these mutual permutations can obtain m different values
will be
r0 + r1u + r2u 2 + · · · + rm−1u m−1 ,
r0, r1, r2 . . . rm−1 being symmetric functions of the n quantities.” 95
89 Actually, HAMILTON thought highly of JERRARD’S results, which he interpreted in a restricted
frame. Although JERRARD’S claim for solving general equations could not be supported, the method
which he had employed was nevertheless of great importance since it — if applied to the quintic —
could reduce it to the normal trinomial form x 5 + px + q = 0.
90 For instance (Cayley, 1861; Cockle, 1862; Cockle, 1863).
91 (N. H. Abel, 1839).
92 (Holmboe in ibid., 409).
93 (N. H. Abel, 1826a, 71, 82). See quotation on page 106.
94 (Holmboe in N. H. Abel, 1839, 411–413).