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

110 Chapter 6. Algebraic insolubility of the quintic
2. ABEL had inherited a result, the CAUCHY-RUFFINI theorem, which limited the
possible numbers of values of rational functions under permutations of their arguments
(see theorem 4). Applied to the quintic, he observed that the result proved that
no function of five quantities could exist which took on three or four different
values when its arguments were interchanged. ABEL proceeded by exploring
the remaining cases, i.e. function of five quantities, which took on two or five
different values.
Although ABEL chose to present his detailed studies of particular cases before linking
these two preliminaries, I have chosen to provide this logical link in the following
section.
6.5 Permutations linked to root extractions
A very central link between the two preliminaries described above was provided to-
ward the end of ABEL’S argument. 31 There, he linked the number of values taken by
a function v under all permutations of its arguments to the minimal degree of a poly-
nomial equation which had v as a root and symmetric functions as coefficients. This
equation is the irreducible equation corresponding to v and was later to take a very
central position in his general theory of solubility (see chapter 8).
ABEL let v designate any rational function of x1, . . . , xn which took on m different
values v1, . . . , vm under permutations of the quantities x1, . . . , xn. By this, he meant
that the function v had the m different formal appearances v1, . . . , vm when its argu-
ments were permuted. Of course, v itself was identical to one of these values but as the
typesetting suggests, ABEL distinguished the values from the function. ABEL formed
the equation
m
∏ (v − vk) =
k=1
m
∑ qkv k=0
k = 0,
and claimed that the coefficients q0, . . . , qm were symmetric functions of the quantities
x1, . . . , xn. ABEL gave no reference and no proof of this assertion, which is now an easy
consequence of one of LAGRANGE’S theorems concerning resolvents (theorem 1).
ABEL also maintained that it was impossible to express v as a root of any equation
of lower degree with symmetric coefficients. He proved this through a reductio ad
absurdum by assuming that
µ
∑ tkv k=0
k = 0 (6.10)
was such an equation where the t k were symmetric, and µ < m. If v1 was a root of
(6.10) it would be possible to divide the polynomial in (6.10) by (v − v1) and obtain
31 (N. H. Abel, 1826a, 81–82)