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

8.3. Refocusing on the equation 175
The central question of this part of the paper was whether k1 = ν, i.e. whether all
the roots of P = 0 were found in the sequence above. ABEL answered this important
question by a nice application of GAUSS’ primitive roots, although his presentation
in the notebook becomes increasingly obscure (see figure 8.1). Eventually, nothing
but a sequence of equations can be found. However, ABEL’S intended argument can
be inferred and reconstructed. In the following, I add some explanation to ABEL’S
equations based on arguments by HOLMBOE and SYLOW. 25
In order to demonstrate that s
1 1
µ µ
1 , . . . , sk
were rational functions of s
1 1 µ , ABEL let m
denote a primitive root of the modulus µ and recast the procedure described above as
1
µ
s1
1
µ
s2
= p0s mα
µ ,
m
= p1s
α
µ
1 ,
1
µ
sk
= pk−1s mα
µ .
At some point, say after the k th iteration, the procedure would stabilize and give
s 1 m
µ = s αk k−1
µ × ∏ p
u=0
muα
k−u−1 .
By the same argument as above, ABEL could write
m αk − 1
µ
.
(8.7)
= integer, (8.8)
and he concluded that k divided µ − 1. This conclusion can be seen to impose a mini-
mality condition upon k with respect to (8.8). However, in ABEL’S equations no men-
tion of such a minimality requirement can be found. The congruence (8.8)
led ABEL to introduce n such that
m αk ≡ 1 (mod µ)
αk = (µ − 1) n.
In subsequent reasoning, ABEL repeatedly used the fact that (k, n) = 1 without going
into details. However, it is a consequence of the minimality of k mentioned above.
Through a sequence of deductions based on primitive roots and congruences inspired
by GAUSS, ABEL could link a number β to the sequence (8.7) such that
βk = µ − 1.
25 (Holmboe in N. H. Abel, 1839, vol. 2, 288–293), (Sylow in N. H. Abel, 1881, vol. 2, 329–338), and (L.
Sylow, 1902, 18–22).