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

5.2. LAGRANGE’s theory of equations 69
had coefficients which depended rationally on the coefficients m and n of the original
quadratic (5.5). 35 This may not be so surprising because today this is easily realized by
observing that the coefficients are symmetric in f [(x) (y)] and f [(y) (x)]. However,
this was precisely the result, which LAGRANGE was about to prove.
Subsequently, LAGRANGE carried out the rather lengthy argument for the general
cubic. Thereby, he proved that the equation which had the six values of f under all
permutations of the three roots of the cubic as its roots would be rationally expressible
in the coefficients of the cubic.
Based on these illustrative cases of equations of low (second and third) degrees,
LAGRANGE could state the following two results as a general theorem generalizing
the argument sketched for the quadratic above. 36 In the general case, the degree of the
equation was denoted µ, and the polynomial having all the values which the given
function f assumes under permutations of the µ roots was denoted Θ and its degree
ϖ. LAGRANGE then stated:
1. The degree ϖ of Θ divides µ! where µ is the degree of the proposed equation,
and
2. The coefficients of the equation Θ = 0 depend rationally on the coefficients of
the original equation.
In his proof of this general theorem, LAGRANGE’S notation and machinery re-
stricted his argument slightly. Because he worked with permutations acting on func-
tions and had no way of clarifying the underlying sets of permutations, his argu-
ments — which contain all the necessary ideas — may seem to rely on analogies with
the cases of low degrees. 37 Be that as it may, by any contemporary standards, LA-
GRANGE’S argument must have been a convincing proof and LAGRANGE’S general
theorem became an immensely important tool in the investigations of future alge-
braists.
“From this it is clear that the number of different functions [i.e. different values
obtained by permuting the arguments] must increase following the products of
natural numbers
1, 1.2, 1.2.3, 1.2.3.4, . . . , 1.2.3.4.5 . . . µ.
Having all these functions one will have the roots of the equation Θ = 0; thus, if
it is represented as
Θ = t ϖ − Mt ϖ−1 + Nt ϖ−2 − Pt ϖ−3 + · · · = 0,
35 (Lagrange, 1770–1771, 361).
36 (ibid., 369–370).
37 Since LAGRANGE’S proof can easily be adapted to newer frameworks of proof, this interpretation
may be a matter of personal taste. However, I do see a major difference between LAGRANGE’S proof
by analogy and pattern and the proof later given by CAUCHY (see below).