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

70 Chapter 5. Towards unsolvable equations
one will have ϖ = 1.2.3.4 . . . µ and the coefficient M will equal the sum of all the
obtained functions, the coefficient N will equal the sum of all products of these
functions multiplied two by two, the coefficient P will equal the sum of all products
of the functions multiplied three by three, and so on. [. . . ]
And since we have demonstrated above that the expression Θ must necessarily
be a rational function of t and the coefficients m, n, p, . . . of the proposed equation,
it follows that the quantities M, N, P, . . . are necessarily rational functions
of m, n, p, . . . which one can find directly as we have seen done in the preceding
sections.” 38
Expressed in modern mathematical language, the first part of the above result is
the equivalent of the Lagrange’s Theorem of group theory, which states that the order
of any subgroup divides the order of the group. As we shall see in section 5.6, the
first general proof was given by A.-L. CAUCHY (1789–1857) based on his approach to
working with permutations.
The second part of the result was used extensively by ABEL, although he never
gave references when applying it. ABEL used the result in a form equivalent to the
following theorem, formulated in a compact notation.
Theorem 1 If φ � �
x1, . . . , xµ is a rational function which takes on the values φ1, . . . , φϖ under
all permutations of its arguments x1, . . . , xµ and the equation
Θ =
ϖ
ϖ
∏ (v − φk) = ∑ Akv k=1
k=0
k
is formed, then all the coefficients A0, . . . , Aϖ are symmetric functions of x1, . . . , xµ. ✷
(5.6)
The link between the above theorem as used by ABEL and LAGRANGE’S second
result can be obtained through a result which I denote Waring’s formulae. These for-
mulae, obtained by NEWTON and WARING by different routes and described in the
next section, were incorporated by LAGRANGE in his work and must have been ac-
cepted as common knowledge in LAGRANGE’S era. As quoted above, LAGRANGE’S
38 “D’où l’on voit clairement que le nombre des fonctions différentes doit croître suivant les produits
des nombres naturels
1, 1.2, 1.2.3, 1.2.3.4, . . . , 1.2.3.4.5 . . . µ.
Ayant toutes ces fonctions on aura donc les racines de l’équation Θ = 0; de sorte que, si on la
représente par
Θ = t ϖ − Mt ϖ−1 + Nt ϖ−2 − Pt ϖ−3 + · · · = 0,
on aura ϖ = 1.2.3.4 . . . µ; et le coefficient M sera égal à la somme de toutes les fonctions trouvées,
le coefficient N égal à la somme de tous les produits de ces fonctions multipliées deux à deux, le
coefficient P égal à la somme de tous les produits des mêmes fonctions multipliées trois à trois, et
ainsi de suite. [. . . ]
Et comme nous avons démontré ci-dessus que l’expression de Θ doit être nécessairement une fonction
rationnelle de t et des coefficients m, n, p, . . . de l’équation proposée, il s’ensuit que les quantités
M, N, P, . . . seront nécessairement des fonctions ratinnelles de m, n, p, . . . qu’on pourra trouver
directement, comme nous l’avons pratiqué dans les Sections précédentes.” (Lagrange, 1770–1771,
369).