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

6.9. Reception of ABEL’s work on the quintic 133
HOLMBOE’S proof implicitly involved LAGRANGE’S notion of semblables functions
(functions which are altered in the same way by the same permutations), and argued
directly that any function of five quantities, which took on five different values, must
have the form of a fourth degree polynomial in which the coefficients were symmetric
functions of x1, . . . , x5.
The final noteworthy contribution by HOLMBOE to ABEL’S impossibility proof was
his calculations relating to the process described as inversion of polynomials. HOLM-
BOE proved — through manipulations on power sums — that any fourth degree poly-
nomial v in x
could be inverted into
v =
x =
4
∑ rαx
α=0
α
4
∑ sαv
α=0
α . 96
The proof is a tour de force dealing with symmetric functions, much in the style of E.
WARING (∼1736–1798), although in a clearer notational setting.
In his commentary, HOLMBOE did not penetrate to the core of the problems spotted
by HAMILTON. Instead, he elaborated many of ABEL’S arguments and manipulations
and supplied proofs of obscure passages. HOLMBOE’S only real reservation against
ABEL’S proof concerned the classification of functions with five values, and HOLM-
BOE provided an alternative deduction using methods and concepts introduced by
LAGRANGE and quite familiar to ABEL.
KÖNIGSBERGER. While HOLMBOE’S elaboration of ABEL’S classification of func-
tions with five quantities might have settled HAMILTON’S unease on this objection,
it took longer before HAMILTON’S other reservation was lifted. The objection raised
against ABEL’S classification of algebraic expressions was lifted in two steps: In 1869,
KÖNIGSBERGER demonstrated how ABEL’S classification could be rescued by modi-
fying the claims concerning the orders and degrees of the coefficients in the represen-
tation
v = q0 + p 1 n + q2p 2 n + · · · + qn−1p n−1
n .
(see page 104). 97 The slight modification which KÖNIGSBERGER introduced revali-
dated ABEL’S hierarchy on algebraic expressions, and showed that ABEL’S “mistake”
was of no real consequence to the proof. KÖNIGSBERGER had been stimulated to make
95 “De la même manière on peut démontrer que, si u signifie une fonction donnée de n quantités qui
prend m valeurs différentes lorsqu’on échange ces n quantités entre elles de toutes les manières
possibles, la forme générale de la fonction de n quantités qui par leurs permutations mutuelles peut
obtenir m valuers différentes sera
r0 + r1u + r2u 2 + · · · + rm−1u m−1 ,
r0, r1, r2 . . . rm−1 étant des fonctions symétriques des n quantités.” (Holmboe in ibid., 413).
97 (Königsberger, 1869).