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

358 Chapter 19. The Paris memoir
The ensuing step was, however, unwarranted as ABEL claimed that
p = Γ (ν + 1) R3 (β)
θ (ν)
.
1 (β)
One can guess how ABEL came to the latter belief by applying the rule of G.-F.-A. DE
L’HOSPITAL (1661–1704) v times to the definition of p as both numerator and denomi-
nator vanish. In the above presentation, the problem which ABEL’S deduction suffered
from is hidden in the notation. First of all, ABEL’S way of suppressing the subscript
k has made the β and x appear symbolically similar, although x is a true variable
whereas β1, . . . , βα are the roots of a certain polynomial. This distinction is at the core
of SYLOW’S objection to ABEL’S argument. 18 However, with a minor adjustment to
the definitions, ABEL’S final product (19.9) of the argument could be allowed.
19.2.3 Main Theorem II
After the first four sections of the Paris memoir, ABEL had thus obtained a formula
which was essentially (apart from the corrections indicated above) the following,
v =
µ
µ �
∑ ψ (xk) = ∑
k=1
k=1
f (x k, y k) dx k = C − Πφ (x) +
α
∑
ν=1
ν dν−1φ1 (x)
dxν−1 .
This expression allowed him to commence a study of the number of free parameters
which would eventually lead to the second main theorem — the celebrated Abelian
Theorem. To follow his argument, we need to backtrack a little to properly understand
the use of the eliminant equation r = 0 (see page 352).
ABEL’S trick was first to study the consequences of one further assumption con-
cerning the factor F0 of r containing the indeterminate quantities. ABEL assumed that
F0 had α distinct zeros,
α
F0 (x) = ∏ (x − βk) k=1
µ k ;
an assumption which — provided F0 is not a constant — introduced α linear interrelations
among the coefficients q0, . . . , qn−1 of the auxiliary polynomial θ (y) (19.3). 19
ABEL found the fact that the coefficients of θ (y) formed a non-independent set to be —
in general — a contraction of the original hypothesis which required nothing of the co-
efficients q0, . . . , qn−1. Consequently, he concluded that F0 (x) had to be a constant and
r (x) could not — in general — contain any factor independent of the auxiliary quanti-
ties.
Under this assumption, ABEL proceeded to describe the various functions involved.
The important outcome of these investigations was that a certain function f2 (x) in-
troduced much earlier in the investigations, reduced to unity, thereby providing the
result that f (x, y) χ ′ (y) equated the entire function f1 (x, y).
18 (Sylow in N. H. Abel, 1881, II, 295).
19 See box 19.2.3.