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

19.2. The contents of ABEL’s Paris result and its proof 369
which was of degree µ − α, has as its roots xα+1, . . . , xµ and whose coefficients were
rational functions of x1, . . . , xα and y1, . . . , yα. Thus, ABEL concluded from Main Theo-
rem I that any sum such as ∑ α k=1 ψ k (x k) could be expressed by a known function v and
a similar sum of functions,
α
∑ ψk (xk) = v −
k=1
µ
∑
k=α+1
ψk (xk) . (19.29)
The relation γ = µ − α. The expression (19.29) at first might seem like a mere rep-
etition of the Main Theorem I, but as ABEL stressed, the number of terms on the right
hand side (µ − α) shows remarkable features. The stress put on the number γ is cer-
tainly one of the important aspects of ABEL’S paper, and it has received widespread
mathematical interest not least after G. F. B. RIEMANN (1826–1866) transformed it
into a coherent concept of genus, see section 19.3. For now, we focus our attention
completely on ABEL’S argument and the inner logic of the paper.
As he had done above, ABEL again divided his argument by whether r has a factor
independent of the indeterminates or not. He started with the latter case, F0 (x) = 1
for which he found that all the coefficient functions q0, . . . , qn−1 were arbitrary and
their hq k + 1 coefficients had to correspond to the indeterminates,
α =
n−1
n
∑ (hqk + 1) = ∑ hqk + n − 1.
k=1
k=1
Obtaining a corresponding formula for the other case, in which F0 (x) �= 1, proved
much more tedious. In general, ABEL claimed, the equation
r (x) = F0 (x) F (x) (19.30)
would impose hF0 conditions, but particular forms for y can eliminate some of these
conditions. 35 If the number of conditions imposed by (19.30) is hF0 − A, the number
of indeterminates could be counted as
α =
n−1
∑ (hqk + 1) − (hF0 − A) . (19.31)
k=1
On the other hand, ABEL easily obtained from the definitions
and
hr = h
hr = hF0 + hF = hF0 + µ
� �
n
n
∏ θ (yk) = ∑ hθ (yk) ,
k=1
k=1
35 Here, a remarkably clear juxtaposition of “in general” and “particular cases” was given.