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

72 Chapter 5. Towards unsolvable equations
from the coefficients of the equation. 42 The solution was the so-called Waring’s Formu-
lae giving a procedure alternative to one given earlier by NEWTON. From this, WAR-
ING proceeded to show how any function of the roots of the form (modern notation
writing Σµ for the symmetric group)
∑ x
σ∈Σµ
a1 σ(1) xa2 . . . xaµ
σ(2) σ(µ) with a1, . . . , aµ non-negative integers (5.7)
could be expressed as an integral function of the power sums of the roots. 43 Thus,
WARING had demonstrated that all rational and symmetric functions of x1, . . . , xµ de-
pended rationally on the power sums and thus on the coefficients of the equation by
the preceding result. 44
Although this important theorem was stated and proved by WARING, it entered
the mathematical toolbox of the early nineteenth century mainly through LAGRANGE’S
adaption of it in his Réflexions (which is the reason for treating it at this place). While
WARING’S notation and letter-manipulating approach had hampered his presentation,
LAGRANGE dealt with it in a clear and integrated fashion in the Réflexions. 45 There, he
observed that if the function f had the form
f
��x ′ ′′
, x � � �
x
′′′�
x iv�
. . .
indicating that x ′ and x ′′ appeared symmetrically, the roots of the equation Θ = 0
(5.6) would be equal in pairs, whereby the degree could be reduced to µ!
2 . After briefly
studying a few other types of functions f , LAGRANGE concluded that if f had the form
f
�
,
��
x ′ , x ′′ , x ′′′ , . . . , x (µ)��
,
i.e. was a symmetric function of the roots, the degree of the equation Θ = 0 (5.6) could
be reduced to one and f would be given rationally in the coefficients of the original
equation.
5.3 Solubility of cyclotomic equations
Thirty years after LAGRANGE’S creative studies on known solutions to low degree
equations, and in particular properties of rational functions under permutations of
their arguments, another great master published a work of profound influence on
early nineteenth century mathematics. In Göttingen, GAUSS was located at a physical
distance from the emerging centers of mathematical research in Paris and Berlin. By
1801, the Parisian mathematicians had for some time been publishing their results in
42 (Waring, 1770, 1–5).
43 (ibid., 9–18).
44 By formal equality, all terms of the same degree would have to have identical coefficients, and thus
any rational symmetric function could be decomposed into functions of the form (5.7).
45 (Lagrange, 1770–1771, 371–372).