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

114 Chapter 6. Algebraic insolubility of the quintic
First, ABEL tacitly applied an equivalent to Waring’s formulae (see section 5.2.4) to
express v rationally in x1 and the elementary symmetric functions A0, . . . , A3 occur-
ring as coefficients in the equation
0 =
5
∏ (x − xk) = x
k=2
4 + A3x 3 + A2x 2 + A1x + A0.
To ABEL, the calculations to obtain this were straightforward and not worth mention-
ing. When ABEL factorized the general quintic as
0 =
5
∏
k=1
(x − x k) = (x − x1)
4
∑
k=0
A kx k =
5
∑ akx k=0
k ,
he found that the coefficients A0, . . . , A4 depended rationally on a0, . . . , a5. Conse-
quently, v could also be expressed rationally in x1 and a0, . . . , a5 as
v = t
φ (x1) ,
where both t and φ (x1) were entire functions of x1, a0, . . . , a5. By inserting the other
roots x2, . . . , x5 for x1 in φ (x1), ABEL obtained another four entire functions in which
the coefficients were symmetric functions of x1, . . . , x5. When ABEL multiplied both
numerator and denominator by ∏ 5 k=2 φ (x k), 35 tacitly used LAGRANGE’S theorem (1)
on resolvents, and reduced the degree according to the relationship imposed by the
quintic equation, he obtained v in the desired form of a fourth degree polynomial in
x1.
Five-valued functions in general. In order to obtain a standard form of all functions
of five quantities having five values, ABEL relied on an extensive investigation of par-
ticular cases. Denoting by v any function of five quantities, which took on the five
values v1, . . . , v5 when all its arguments were permuted, ABEL introduced an inde-
terminate m and formed the function x m 1 v. When only x2, . . . , x5 were permuted, this
function would attain its values from the list
x m 1 v1, . . . , x m 1 v5. (6.12)
ABEL let µ denote the number of different values of x m 1 v when x2, . . . , x5 were per-
muted in all possible ways. He then considered the different cases corresponding to
different values of µ in detail and either eliminated them through a reductio ad absur-
dum or reduced them to the standard form (6.11). Throughout this procedure, it is
important to keep in mind which quantities are permuted, and ABEL was not always
very explicit.
35 A similar argument resembling multiplying the denominator by its conjugate is described in section
6.3.2.