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

100 Chapter 6. Algebraic insolubility of the quintic
This shift from the trial-and-error based search for solutions toward a theoretical
and general investigation of the class of algebraic expressions marks ABEL’S first break
with the traditional approach to the theory of equations. ABEL investigated the extent
to which algebraic expressions could satisfy given polynomial equations and was led
to describe necessary conditions. By this choice of focus, ABEL implicitly introduced
a new object, algebraic expression, into the realm of algebra, and the first part of his
paper can be seen as an opening study of this object, devised in order to obtain a firm
description of it and to prove the first central theorem concerning it. 12 In section 19.3,
another aspect of ABEL’S concept of algebraic expression is taken up.
6.2 Outline of ABEL’s proof
The paper in CRELLE’S Journal für die reine und angewandte Mathematik can be divided
into four sections reflecting the overall structure of ABEL’S proof. In the first section,
ABEL introduced his definition of algebraic functions and classified these by their or-
ders and degrees. He used this definition to study the restrictions imposed on the
form of algebraic expressions if they had to be solutions to a given solvable equation.
In doing so, he proved the result — which RUFFINI had failed to see — that any radical
(algebraic sub-expression) contained in a supposed solution would depend rationally
on the roots of the equation (see section 6.3).
In the second section, ABEL reproduced the elements of A.-L. CAUCHY’S (1789–
1857) theory of permutations from 1815 needed for his proof. 13 These included CAUCHY’S
notation and the result described above as the CAUCHY-RUFFINI theorem (section 5.6)
demonstrating that no function of the five roots of the general quintic could take on
three or four different values under permutations of these roots (see section 6.4).
The third part contained detailed and highly explicit investigations of functions
of five quantities taking on two or five different values under all permutations of the
roots. Through an explicit theorem, which linked the number of values under permu-
tations to the degree of the root extraction (see section 6.5), ABEL demonstrated that
all non-symmetric rational functions of five quantities could be reduced to two basic
forms.
Finally, these preliminary sections were combined to provide ABEL’S impossibil-
ity proof by discarding each of a number of cases ad absurdum (section 6.6). ABEL’S
argument can be outlined in the following steps:
1. ABEL introduced a classification of algebraic expressions to obtain a standard
form, rational in the roots, which all possible solutions to the general quintic
equation had to possess.
12 Studying algebraic expressions as objects has been seen as a first step in what later became the introduction
of functions as mappings (especially automorphisms) into algebra and separating functions
from their ties with analysis. (Kiernan, 1971, 70)
13 (A.-L. Cauchy, 1815a).