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

52 Chapter 4. The position and role of ABEL’s works within the discipline of algebra
ABEL planned to apply this theory to the division problems for circular and elliptic
functions. However, only his reworking of GAUSS’ study of cyclotomic equations was
published in the paper.
Together, the insolubility proof and the study of Abelian equations can be inter-
preted as an investigation of the extension of the concept of algebraic solubility. On
one hand, the insolubility proof provided a negative result which limited this exten-
sion by establishing the existence of certain equations in its complement. On the other
hand, the Abelian equations fell within the extension of the concept of algebraic solu-
bility and thus ensured a certain power (or volume) of the concept.
Grand Theory of Solubility. In a notebook manuscript — first published 1839 in the
first edition of ABEL’S Œuvres — ABEL pursued further investigations of the exten-
sion of the concept of algebraic solubility. In the introduction to the manuscript, he
proposed to search for methods of deciding whether or not a given equation was solv-
able by radicals. The realization of this program would, thus, have amounted to a
complete characterization of the concept of algebraic solubility.
ABEL’S own approach to this program was based upon his concept of irreducible
equations. In the first part of the manuscript — which appears virtually ready for the
press — ABEL gave his definition of this concept. Arguing from the definition, he
proved some basic and important theorems concerning irreducible equations.
In the latter part of the manuscript — which is less lucid and toward the end con-
sists of nothing but equations — ABEL reduced the study of algebraic expressions sat-
isfying a given equation of degree µ to the study of algebraic expressions which could
satisfy an irreducible Abelian equation whose degree divided µ − 1. However, ABEL’S
researches were inconclusive. When ABEL’S attempt at a general theory of algebraic
solubility eventually was published in 1839, another major player in the field, GA-
LOIS, had also worked on the subject. Inspired by the same tradition and exemplary
problems as ABEL had been, GALOIS put forth a very general theory with the help of
which the solubility of any equation could — at least in principle — be decided.
GALOIS’ writings were inaccessible to the mathematical community until the mid-
dle of the nineteenth century. His style was brief and — at times — obscure and unrig-
orous. Many mathematicians of the second half of the nineteenth century — starting
with J. LIOUVILLE (1809–1882) who first published GALOIS’ manuscripts in 1846 —
invested large efforts in clarifying, elaborating, and extending GALOIS’ ideas. In the
process, the theory of equations finally emerged in its modern form as a fertile subfield
of modern algebra. Part of this evolution concerned mathematical styles. The highly
computational mathematical style of the eighteenth century, to which ABEL had also
adhered, was superseded. The old style had been marked by lengthy, rather concrete,
and painstaking algebraic manipulations. In the nineteenth century, this was replaced
by a more conceptual reasoning, early glimpses of which can be seen in ABEL’S works
on the algebraic solubility of equations.