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

318 Chapter 16. The idea of inverting elliptic integrals
i.e.
φ
� � �
2mαω nω
= φ
4v + 1 4v + 1
�
+ tω = (−1) t � �
nω
φ
4v + 1
which for n = 1 made φ � �
ω
4n+1 accessible (known).
Throughout, ABEL had explicitly only considered the division into a prime number
of parts. When ABEL then made the further assumption that 4v + 1 = 1 + 2 n , he found
by considering the expressions obtained that all the root extractions reduced to square
roots. In particular, ABEL had to use that the solution of the equation θ2n−1 = 1 could
be reduced to square roots; this result is precisely the main result of GAUSS’ research
on the division of the circle. Combining this result with the case of bisection and
the general integral multiplication, ABEL could summarize his investigations on the
division of the lemniscate:
“The value of the function φ � �
mω can be expressed by square roots whenever
n
n is a number of the form 2n or a prime number of the form 1 + 2n or a product of
multiple numbers of these two forms.” 37
Two aspects of ABEL’S result merit attention. First, ABEL’S argument hinges on
the factors 2 m0, 2 n 1 + 1, . . . , 2 n k + 1 of n to be relatively prime because he wanted to
decompose any number m ′ into its residues modulo these factors,
�
m0 m1
φ +
2n0 2n1 + 1 + · · · + mk 2n � �
= φ
k + 1
m ′
2 n0 (2 n 1 + 1) . . . (2 n k + 1)
If two of the Fermat primes were identical, the decomposition would no longer be pos-
sible. 38 ABEL’S deductions immediately leading to the stated theorem contain tacitly
the distinctness of the Fermat primes, but it could have been explicitly included in the
statement.
Second, the result states a sufficient condition of geometrical constructibility and
says absolutely nothing of the necessity of this condition. The same can be said of
GAUSS’ stated result on the division of the circle. However, GAUSS also stated that the
division of the circle would lead to precisely his equations and would therefore not
be possible with ruler and compass unless the number of parts were of the prescribed
form. Similarly, ABEL could have stated that division of the lemniscate was only pos-
sible if n was a product of a power of 2 and distinct Fermat primes. 39 However, the
proof of such a statement would go beyond the types of questions which ABEL asked
concerning these classes of equations.
37 “La valeur de la fonction φ � �
mω
n peut être exprimée par des racines carrées toutes les fois que n
est un nombre de la forme 2n ou un nombre premier de la forme 1 + 2n , ou même un produit de
plusieurs nombres de ces deux formes.” (N. H. Abel, 1828b, 168).
38 Consider writing e.g. 3
39
a+b
25 as 5 with a, b ∈ Z.
For a proof using more modern techniques, see (M. Rosen, 1981).
�
.