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

19.2. The contents of ABEL’s Paris result and its proof 359
Linear interdependence of q0, . . . , qn−1 In order to see how ABEL obtained the linear
interrelations among the coefficients of θ (y), we notice from combining the factoriza-
tion (19.5) with the definition of r (19.4),
r (x) = F (x) · F0 (x)
=
n
∏ θ
k=1
�
y (k) (x)
�
=
n n−1
∏ ∑
k=1 m=0
�
qm (x) y (k) �m (x) .
Thus, since r (β1) = F (β1) · 0 = 0, some k must exist for which θ
n−1 �
∑ qm (x) y
m=0
(k) �m (x) = 0.
�
y (k) �
(β1) = 0, i.e.
This relation is a linear interdependence among the q0, . . . , qn−1 in which the
here serving as coefficients, are functions of x.
Box 10: Linear interdependence of q0, . . . , qn−1
�
y (k)� m
,
ABEL’S way of proving f (x, y) χ ′ (y) to be an entire function. ABEL’S investiga-
tions leading to the result that f (x, y) χ ′ (y) was an entire function progressed along
complicated and tedious arguments. At the very outset of the paper, ABEL had split
the rational function of x in the following way
f (x, y) χ ′ (y) = f1 (x, y)
, (19.16)
f2 (x)
where f2 (x) was an entire function of x independent of y. By combining this with the
important partial differentiation (19.7), ABEL found
f (x, y) dx =
f1 (x, y)
f2 (x) · χ ′ (y) dx
1
= −
F0 (x) · F ′ (x) · f2 (x)
n
∑
k=1
f1 (x, y k)
χ ′ (y k)
r
θ (y k) ∂θ (y k) .
Later, during his investigations leading to the Main Theorem I, ABEL had studied the
function
which he had chosen to write as
f1 (x, y)
f1 (x, y)
r
∂θ (y)
θ (y)
r
θ (y) ∂θ (y) = R′ (y) + R (x) y n−1 ,
where R ′ (y) indicated an entire function of x and y in which no powers of y beyond
the (n − 2)’nd occur, and R (x) was an entire function of x independent of y. By use of