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

106 Chapter 6. Algebraic insolubility of the quintic
established. C. SKAU considers the Euclidean algorithm among the central pillars of
ABEL’S impossibility proof. 22 In section 7.3.1, I illustrate how it, indeed, — together
with the concept of irreducibility — played an important role in ABEL’S theory of
equations.
In the very same paragraph, ABEL let
µ
∑ tuz
u=0
u = 0 (6.8)
denote the factor of (6.7) of lowest degree with rational coefficients and continued with
the following statement implicitly introducing irreducibility which ABEL had not used
or defined thus far:
and let
“Let that equation [here (6.7)] be
s0 + s1z + s2z 2 · · · + s k−1z k−1 + z k = 0
t0 + t1z + t2z 2 · · · + tµ−1z µ−1 + z µ
be a factor of its first term [left hand side], where t0, t1 etc. are rational functions
of p, r0, r1 . . . rn−1; then also
t0 + t1z + t2z 2 · · · + tµ−1z µ−1 + z µ = 0
and it is clear, that it can be assumed to be impossible to find an equation of the
same form of lower degree.” 23
Thus, certain roots of (6.7) would also be roots of (6.8), ABEL argued, and the µ
roots of (6.8) would also be roots of z n − p = 0. In the case µ = 1, it would be easy to
write z, i.e. p 1 n , as a rational function of t0 and t1, and thereby as a rational function of
p, r0, . . . , rn−1 from (6.6), contrary to the assumption imposed by theorem 2.
Since µ ≥ 2, ABEL let z and αz denote two distinct common roots of (6.8) and
z n − p = 0. When he inserted them into (6.8), he obtained
22 (Skau, 1990, 54).
23 “Die Gleichung sei
und
µ−1
∑ tu (α
u=0
u − α µ ) z u = 0 (6.9)
s0 + s1z + s2z 2 · · · + s k−1z k−1 + z k = 0
t0 + t1z + t2z 2 · · · + tµ−1z µ−1 + z µ
ein Factor ihres ersten Gliedes, wo t0, t1 etc. rationale Functionen von p, r0, r1 . . . rn−1 sind, so ist
auch
t0 + t1z + t2z 2 · · · + tµ−1z µ−1 + z µ = 0
und es ist klar, daß man es als unmöglich annehmen kann, eine Gleichung von niedrigerem Grade
von der nemlichen Form zu finden.” (N. H. Abel, 1826a, 71).