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

19.2. The contents of ABEL’s Paris result and its proof 363
Later, in chapter 21, we shall have more to say on this notion of ‘in general, except for
certain particular cases.’
Writing f1 (x, y), which was by construction an entire function of y, in the way
ABEL concluded
Based on the identity
this evolved into
f1 (x, y) =
n−1
∑ tm (x) y
m=0
m ,
htmy m k < hχ′ (y k) − 1 for k = 1, . . . , n and m = 0, . . . , n − 1.
h (tmy m ) = ht k + mhy,
htm + mhy < hχ ′ (y k) − 1.
ABEL now combined the information contained in the equations () obtaining
hχ ′ (y k) − mhy k − 1 = (n − m − k) hy k +
k−1
∑ hyu − 1.
u=1
Since y1, . . . , yn were assumed to be ordered according to decreasing degrees, the min-
imal value among these (over k = 1, . . . , n) was obtained for k = n − m, resulting in
the value
� ′
min hχ (yk) − mhyk − 1
k=1,...,n
� = hχ ′ n−m−1
(yn−m) − mhyn−m − 1 =
Therefore, because htm was an integer,
where 0 < εn−m−1 ≤ 1.
htm =
∑
u=1
hyu − 1.
n−m−1
∑ hyu − 2 + εn−m−1, (19.20)
u=1
Grouping of roots according to their degree. The next step in ABEL’S analysis was
to write
from which he obtained
hy1 = m1
µ1
with (m1, µ1) = 1
hy1 = hy2 = · · · = hyµ 1 = m1
by an argument involving tools from the theory of equations. In his investigations
on algebraic solubility of equations, ABEL had proved (see e.g. chapter 8) that if an
equation (here χ (y) = 0) was satisfied by an expression such as y = Ax
µ1
m1 µ 1 , an entire