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

19.2. The contents of ABEL’s Paris result and its proof 365
and thus
ε kτ+β = � K ′ − K � + ζ
which is a contradiction. Thus K = K ′ and A τ,β is the smallest number such that
(19.22) holds. This condition of minimality, which ABEL just noticed as a matter of
µτ
> 1
fact, was soon (see below) invoked and found to be of great use.
found
When ABEL spelled out the results obtained above for the first sequence τ = 1, he
ht n−β−1 = −2 + β m1
µ1
+ A 1,β
µ1
= −2 + βm1 + A1,β .
µ1
For ‘small’ β (starting with β = 0), the right hand side is obviously negative, because
A 1,β < µ1 by the definition of ε kτ+β
A 1,β
µ1
= ε β ≤ 1
and the other term vanishes for β = 0. Consequently, some β ′ ≥ 0 existed such that
which obviously meant that
ht n−β−1 < 0 for β = 0, . . . β ′
(19.23)
t n−β−1 = 0 for β = 0, . . . , β ′ . (19.24)
The general form of the function f1 (x, y) in the light of these results thus became
f1 (x, y) =
n−β ′ −1
∑ tk (x) y
k=0
k
in which β ′ was the largest integer less than µ 1
m + 1,
1
� �
µ1
β ′ =
m1
+ 1
.
(19.25)
ABEL’S way of obtaining this ultimate description of β ′ was found by SYLOW to miss
certain particular cases. 28
Based on the expression (19.25) which ABEL had obtained for the function f1 (x, y),
he concluded:
“A function like f1 (x, y) always exists when β does not surpass n − 1.” 29
It is difficult to see exactly what ABEL meant by this phrase, which seems to infer
that the existence of the function was deduced from the representation (19.25). How-
ever, on a logical basis, the existence of the function f1 (x, y) had been presupposed in
the decomposition of f (x, y) χ ′ (y), see (19.16).
28 (Sylow in N. H. Abel, 1881, II, 298).
29 “Une fonction telle que f1 (x, y) existe donc toujours à moins que β ′ ne surpasse n − 1.” (N. H. Abel,
[1826] 1841, 166).