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

142 Chapter 7. Particular classes of solvable equations
7.1 Solubility of Abelian equations
The structure of ABEL’S Mémoire sur une classe particulière 2 is a descent from the gen-
eral to the particular. At the outset, ABEL proposed to study irreducible equations in
which one of the roots depended rationally on another one. The concept of irreducible
equations took a central place in this research (see section 7.3). Part of the study was
especially devoted to circular functions to which ABEL had been led by C. F. GAUSS’
(1777–1855) work on the cyclotomic equation. Besides this application to circular func-
tions, ABEL also worked on another application of the general theory to the division
problem for elliptic functions. Likewise inspired by GAUSS’ Disquisitiones arithmeti-
cae (see section 7.2), this application was, however, not contained in the paper but
had been presented the previous year in a paper on elliptic functions. ABEL was led
by these two applications to an even more general result — valid for a broader class
of equations having rationally related roots. In this section, I outline ABEL’S results
before turning to discussions of his inspirations and methods.
7.1.1 Decomposition of the equation into lower degrees
Throughout the paper, ABEL studied polynomial equations of degree µ,
φ (x) = 0,
in which two roots x1, x ′ were related by the rational function θ,
x ′ = θ (x1) .
The quantities which ABEL considered known in his deductions comprise all coeffi-
cients occurring in φ or θ. From a modern perspective, it will become clear that he also
considered any required roots of unity to be known.
ABEL defined the equation φ (x) = 0 to be irreducible when none of its roots could
be expressed by a similar equation of lower degree (see section 7.3).
Employing the Euclidean division algorithm (see section 7.3) and the notation
θ k (x1) for the k th iterated application of the rational function θ to x1, ABEL found
that the set of roots of φ (x) = 0 split into sequences (chains). He deduced — using the
irreducibility of φ (x) = 0 — that because the two roots x1, x ′ of the equation φ (x) = 0
were rationally related, every iteration θ k (x1) would also be a root of φ (x) = 0. There-
fore, the entire set of roots of φ (x) = 0 could be collected in sequences of equal length,
say n, and he wrote the roots as (µ = m × n), 3
θ k (xu) for 0 ≤ k ≤ n − 1 and 1 ≤ u ≤ m. (7.1)
2 (N. H. Abel, 1829c).
3 For brevity, I have added to ABEL’S notation the convention θ 0 (x1) = x1.