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

152 Chapter 7. Particular classes of solvable equations
1. A general study of equations in which one root depended rationally on another.
2. A restriction to irreducible equations and an application of the concept of ir-
reducibility to prove that if x and θ (x) were roots of the irreducible equation
φ (x) = 0, then so was θ k (x) for all integers k.
3. A study of equations of degree µ = m × n in which the result was obtained
that the solution of such equations could be reduced to solving m algebraically
solvable equations of degree n and a single (generally unsolvable) equation of
degree m.
4. An application of these — and other — results to the class of Abelian equations,
and a demonstration that these were always solvable by radicals.
5. A further application of this result to the circular functions by which GAUSS’
results on the cyclotomic equation were reproduced.
ABEL had further ideas for applications of this new theory to elliptic functions,
but these were not printed on this occasion (see below). In his research on Abelian
equations, KRONECKER much later came to the conclusion that “these general Abelian
equations in reality are nothing but cyclotomic equations.” 18 ABEL’S paper contains,
however, more than just the solubility-result for Abelian equations, and the general
theory of the class of equations with rationally dependent roots sprung from — and
had quite interesting implications for — ABEL’S approach to the theory of elliptic func-
tions.
7.2 Elliptic functions
In his very first publication on elliptic functions entitled Recherches sur les fonctions
elliptiques, 19 ABEL made several interesting innovations. 20 ABEL devoted a large por-
tion of the first part of the Recherches to the inversion of elliptic integrals into elliptic
functions, the extension of these functions into the complex domain, and the study
of algebraic relations involving these functions. He derived addition formulae and
studied the singularities of elliptic functions in order to address the central problem,
which can be summarized in the following way:
Problem 1 (Division Problem) Given an integer m and the value φ (mβ) of an elliptic
function of the first kind, φ, at mβ, express φ (β) by radicals. ✷
18 “[. . . ] so daß dise allgemeinen Abelschen Gleichungen im Wesentlichen nichts Anderes sind, als
Kreistheilungs-Gleichungen.” (Kronecker, 1853, 11).
19 (N. H. Abel, 1827b).
20 The history of these elliptic functions and ABEL’S works on them is studied in much greater depth
in part IV. For the present discussion, I am only concerned with the ideas behind ABEL’S result on
the solubility of Abelian equations.