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

316 Chapter 16. The idea of inverting elliptic integrals
In light of the previous result, ABEL’S investigations mainly concerned the division
of the complete integral into an odd number of equal segments. He did so by first
carrying out the division of the complete integral into 4v + 1 parts. ABEL’S argument
was explicitly designed to make the case accessible with the Gaussian approach to the
solution of cyclotomic equations. A brief outline of ABEL’S reasoning will provide a
few interesting comparisons with GAUSS’ approach and the more general solution of
the problem found in ABEL’S paper on Abelian equations.
ABEL first assumed that 4v + 1 was the sum of two squares, 4v + 1 = α 2 + β 2 and
found that α + β had to be an odd integer. In this case, he found an equation which he
wrote as
φ ((α + βi) δ) = x T
S
(16.12)
with α = mδ, β = µδ, x = φ (δ), and T and S two entire functions of x4 . ABEL’S
real objective was the considerations pertaining to δ = ω
�
α+βi for which he obviously
found that x = φ (δ) = φ was a root of the equation T = 0. It thus became his
� ω
α+βi
objective to solve this equation.
Expressing the roots of T = 0. First, by his very powerful determination of the roots
of φ (ξ) = 0, ABEL found that all the roots of T = 0 were related by
(α + βi) δ = mω + µ ¯ωi = (m + µi) ω
since ω = ¯ω for the lemniscate integral. Thus, any root was contained in the formula
�
m + µi
x = φ
α + βi ω
�
if m and µ were allowed to assume all integral values. However, in order to count
each root only once, ABEL found that the set of roots of T = 0 could be listed as
� �
ρω
φ
α + βi
for − α2 + β 2 − 1
2
≤ ρ ≤ α2 + β 2 − 1
2
(16.13)
and he argued by an application of the Euclidean algorithm for integers: ABEL let λ, λ ′
be determined by the equation
and t denote an integer in order to obtain
αλ ′ − βλ = 1,
µ + β (µλ + tα) −α
� �� �
=k
� µλ ′ + tβ �
� �� �
=k ′
= 0.
Then, with ρ = m + αk − βk ′ , ABEL obtained
m + µi
α + βi
= ρ
α + βi − k − k′ i