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

7.1. Solubility of Abelian equations 151
1) The division of the whole circle into n − 1 parts (n − 1 because the irreducible
= 0),
equation in GAUSS’ research was xn −1
x−1
2) The division into n − 1 parts of another arc which could be constructed after step
1 had been completed, and
3) The extraction of a square root.
The final step, the extraction of a square root, could be assumed to equal the construc-
tion of √ n, GAUSS claimed without providing any proof. Later, ABEL adopted and
proved the assertion.
In the fifth section of the Mémoire sur une classe particulière, ABEL applied his theory
directly to the cyclotomic equation and the circular functions related to the division of
the circle. By the addition formulae for cosine, ABEL could express cos ma rationally
in cos a, and assuming θ (cos a) = cos ma and θ1 (cos a) = cos m ′ a, he obtained
θθ1 (x) = θ � cos m ′ a � = cos � mm ′ a �
= cos � m ′ ma � = θ1 (cos ma) = θ1θ (x) .
From a previously established result (here theorem 7), ABEL found that cos 2π µ could
be determined algebraically — which was a well known result.
ABEL, however, did not stop his investigations of the circular functions at this
point, as he might have done had he only been interested in the algebraic solubility of
the division. Assuming that µ = 2n + 1 was prime, ABEL studied the equation
and used the rational dependency established above
to write
n �
∏ X − cos
k=1
2kπ
�
= 0, (7.14)
2n + 1
θ (cos a) = cos ma
θ k (cos a) = cos m k a.
By an argument based on GAUSS’ primitive roots of the module 2n + 1, ABEL demon-
strated that the roots of (7.14) were
x, θ (x) , θ 2 (x) , . . . , θ n−1 (x) where θ n (x) = x.
Therefore, the equation (7.14) was algebraically solvable by ABEL’S third theorem
(here theorem 6), and ABEL adapted theorem 8 to this particular equation, obtain-
ing the same result as GAUSS had found. Furthermore, ABEL presented a proof of
the result, which GAUSS had only announced, that the square root extracted in step 3
could always be made to equal √ 2n + 1 (in ABEL’S variables).
The contents of ABEL’S Mémoire sur une classe particulière can be summarized in the
following five points depicting a descent from the general to the particular: