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

108 Chapter 6. Algebraic insolubility of the quintic
ensured ABEL that any expression which he was to encounter in the hierarchy of a
solvable equation, would depend rationally on the roots of the given equation.
6.4 ABEL and the theory of permutations: the
CAUCHY-RUFFINI theorem revisited
The second preliminary pillar of ABEL’S impossibility proof was made up of his stud-
ies of permutations and his proof of the CAUCHY-RUFFINI theorem describing the
possible numbers of values of rational functions under permutations of their argu-
ments. Prior to giving his proof of this central result, ABEL summarized much of what
CAUCHY had done in his 1815-paper, 27 and in doing so ABEL took over CAUCHY’S
notation and much of his terminology. But while CAUCHY had begun the process of
liberating the substitutions from the expressions on which they acted, ABEL contin-
ued the tradition of LAGRANGE. Although he occasionally spoke of the “substitution”
[Vervandlung] as an independent object, all his deductions concerned their actions on
expressions.
“Now let
A1
A2
� �
αβγδ . . .
v
abcd . . .
be the value, which an arbitrary function v takes, when therein xa, xb, xc, xd etc. are
inserted instead of xα, xβ, xγ, xδ etc.; then it is clear that, when by A1, A2 . . . Aµ one
denotes the different forms which 1, 2, 3, 4 . . . n can possibly take by interchanges
of the exponents 1, 2, 3 . . . n, the different values of v can be expressed as
� � � � � � � �
A1 A1 A1 A1
v , v , v . . . v .” 28
A3
With this notation, ABEL proved LAGRANGE’S theorem that the number of differ-
ent values of the function v would be a divisor of n!. Next, he introduced the concept
of recurring substitutions [wiederkehrende Verwandlungen] of order p, thereby re-
placing the word degree chosen by CAUCHY. In the 1840s, CAUCHY was to take up
ABEL’S terminology on this point. 29 Through a counting argument based on what
27 (A.-L. Cauchy, 1815a).
28 “Nun sei
� �
αβγδ . . .
v
abcd . . .
der Werth, welchen eine beliebige Function v bekommt, wenn man darin xa, x b, xc, x d etc. statt
xα, x β, xγ, x δ etc. setzt, so ist klar, daß wenn man durch A1, A2 . . . Aµ die verschiedenen Formen bezeichnet,
deren 1, 2, 3, 4 . . . n durch Verwechselung der Zeiger 1, 2, 3 . . . n fähig ist, die verschiedenen
Werthe von v durch
v
� A1
A1
�
, v
� A1
A2
�
, v
� A1
ausgedrückt werden können.” (N. H. Abel, 1826a, 74).
29 (Wussing, 1969, 67).
A3
�
. . . v
Aµ
� A1
Aµ
�