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

68 Chapter 5. Towards unsolvable equations
For example, x ′ x ′′ + x ′′′ would remain unaltered by interchanging x ′ and x ′′ , but any
permutation involving x ′′′ would alter it. Finally, if the function was symmetric (i.e.
formally invariant under all permutations of x ′ , x ′′ , x ′′′ ), he wrote
f �� x ′ , x ′′ , x ′′′�� .
The most important examples of such functions were the elementary symmetric func-
tions,
x ′ + x ′′ + x ′′′ ,
x ′ x ′′ + x ′ x ′′′ + x ′′ x ′′′ , and
x ′ x ′′ x ′′′ .
With this notation and his concept of formal equality, LAGRANGE derived far-
reaching results on the number of (formally) different values which rational func-
tions could assume under all permutations of the roots. With the hindsight that the
set of permutations form an example of an abstract group, a permutation group, one
may see that LAGRANGE was certainly involved in the early evolution of permutation
group theory. As we shall see in the following section, he was led by this approach
to Lagrange’s Theorem, which in modern terminology expresses that the order of a sub-
group divides the order of the group. However, since LAGRANGE dealt with the ac-
tions of permutations on rational functions, he was conceptually still quite far from the
concept of groups. LAGRANGE’S contribution to the later field of group theory laid in
providing the link between the theory of equations and permutations which in turn
led to the study of permutation groups from which (in conjunction with other sources)
the abstract group concept was distilled. 34 More importantly, LAGRANGE’S idea of in-
troducing permutations into the theory of equations provided subsequent generations
with a powerful tool.
5.2.3 LAGRANGE’s resolvents
Another result found by LAGRANGE, of which ABEL later made eminent and frequent
use in his investigations, concerned the polynomial having as its roots all the different
values which a given function took when its arguments were permuted. Starting with
the case of the quadratic equation having as roots x and y
z 2 + mz + n = 0, (5.5)
LAGRANGE studied the values f [(x) (y)] and f [(y) (x)] which were all the values a
rational function f could obtain under permutations of x and y. He then demonstrated
that the equation in t
34 (Wussing, 1969).
Θ = [t − f [(x) (y)]] × [t − f [(y) (x)]] = 0