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

96 Chapter 5. Towards unsolvable equations
spelled out some of the most important algebraic tools of the early 19 th century; there-
fore some of his tools are briefly sketched in the present context. During the proof,
GAUSS dealt with results such as the Euclidean algorithm applied to polynomials and
the “elementariness” of the elementary symmetric functions, both of which will be-
come immensely important in ABEL’S theory of algebraic solubility as described in
subsequent chapters. Whether ABEL studied any of GAUSS’ proofs of the fundamen-
tal theorem of algebra is not clear; there are no explicit references to these proofs in
ABEL’S writings, nor is ABEL anywhere concerned with the existence of roots. 102 Thus,
the similarity of methods in GAUSS’ proof and ABEL’S subsequent algebraic research
may equally well be attributed to their belonging to the same common framework and
mathematical tradition.
Explicitly stressing the connection to the procedure used to determine the greatest
common divisor of integers, GAUSS applied the Euclidean algorithm to polynomials.
Besides producing the greatest common divisor, the procedure also proved that two
polynomials Y, Y ′ have no (non-trivial) common divisor if and only if there exists an-
other pair of polynomials Z, Z ′ such that
ZY + Z ′ Y ′ = 1.
The second tool which GAUSS introduced concerned symmetric functions, and
amounts to the central theorem on symmetric functions. By firstly decomposing any
symmetric function of a, b, c, . . . in a sum of terms
Ma α b β c γ . . .
and secondly imposing an ordering on such terms, GAUSS was able to prove that any
symmetric function could be realized as an entire function of the elementary symmet-
ric functions.
Besides these tools, GAUSS’ argument rested upon central properties of the quan-
tity which he termed the determinant (today called the discriminant) of Y (x) = ∏ (x − x k),
∏ i�=j
� �
xi − xj .
GAUSS was able to demonstrate that the determinant vanishes if and only if Y and
d
dx Y have a common divisor, i.e. a common root.
102 Without references, KLINE writes as if ABEL had given a proof of the fundamental theorem of algebra
(Kline, 1990, 599). I have not been able to identify such a proof, nor have I any idea how KLINE
had come to believe that ABEL had even worked on it.