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

362 Chapter 19. The Paris memoir
Thus, for the individual terms of the sum, which were in general just algebraic func-
tions of x, the ‘degree’ hR needed not be an integer, whence the conclusion
h f1 (x, y k)
χ ′ (y k)
< −1. (19.19)
The investigation now turned to algebraic manipulations of these new symbols.
Determination of the most general form of the function f1 (x, y). ABEL put his new
tool to immediate use. From the general formula
h R1
R2
he derived from (19.19) the inequalities
= hR1 − hR2,
h f1 (x, y k) < hχ ′ (y k) − 1,
which he claimed made “it easy to deduce the most general form of the function
f1 (x, y) in each particular case.” 24 ABEL’S argument — but not its result — has been
found unrigorous at this point, see e.g. SYLOW’S notes, the paper by ELLIOT, and be-
low. 25 However, it is worth following the steps of his argument to see how he went
about it.
Because
ABEL obtained
χ ′ (yk) = ∏ (yk − ym) ,
m�=k
hχ ′ (yk) = ∑ h (yk − ym) ,
m�=k
and when the y1, . . . , y k were ordered according to decreasing degrees,
hy k ≥ hym if k ≤ m,
he found “in general, except for certain particular cases which he did not consider:” 26
h (y k − ym) = hy min(k,m).
The analogy with the ordinary degree operator makes the above-mentioned par-
ticular cases easy to illustrate. For instance, if we have two monic polynomials of the
same degree, the degree of their difference is strictly less than either of the original
degrees,
��
deg x 2 � �
+ x − 1 − x 2 ��
− x + 2 = deg (2x − 3) = 1.
24 “De ces inégalités on déduira facilement dans chaque cas particulier la forme la plus générale de la
fonction f1 (x, y).” (N. H. Abel, [1826] 1841, 162).
25 (Sylow in N. H. Abel, 1881, II, 296–297) and (Elliot, 1876, 404–406).
26 “Alors on aura, en général, excepté quelques cas particuliers que je me dispense de considérer:”
(N. H. Abel, [1826] 1841, 162).