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

6.3. Classification of algebraic expressions 107
which was an equation of degree at most µ − 1 having some of the roots of the irre-
ducible (6.8) as its roots. In this connection, ABEL actually used the word “irreducible”
for the first time (see the quotation below). Consequently, the polynomial of (6.9)
would have to be the zero polynomial and a contraction had been reached:
“But since the equation z µ + tµ−1z µ−1 · · · = 0 is irreducible, it must, since it is
of the µ − 1’st degree give
which is impossible.” 24
α µ − 1 = 0, α − α µ = 0 . . . α µ−1 − α µ = 0;
The contradicted assumption was that at least one coefficient among r0, . . . , rn−1
was non-zero, and thus the result (lemma 1) had been demonstrated.
When ABEL considered n different values y1, . . . , yn of y resulting from substitut-
ing α k p 1 n for p 1 n in the expression (6.4) for y, he found that these all constituted roots
of the equation when it was assumed to be algebraically solvable. Through labori-
ous, albeit not very difficult, algebraic manipulations including a tacit application of
LAGRANGE’S theorem 1 on resolvents, ABEL then demonstrated that if the equation
was solvable, the coefficients q0, q2, . . . , qn−1 as well as p 1 n would all depend rationally
on these roots (and certain roots of unity, such as α). Thereby, he demonstrated that
all components of a top-level algebraic expression solving a solvable equation were ra-
tional functions of the equation’s roots. By considering any of these components and
working downward in the hierarchy, ABEL demonstrated that this applied equally
well to any component involved in the solution. Thus, he had proved the following
explicitly formulated and very important auxiliary theorem, corresponding to RUF-
FINI’S open hypothesis. 25
Theorem 3 “When an equation can be solved algebraically, it is always possible to give to
the root [solution] such a form that all the algebraic functions of which it is composed can be
expressed by rational functions of the roots of the given equation.” 26
The study of algebraic expressions which ABEL had conducted as a preliminary
to his impossibility proof had produced two central results for the proof. Firstly, it
had provided a hierarchy on the algebraic expressions based on the nesting of root
extractions. Secondly, it had resulted in the auxiliary theorem stated just above, which
24 “Da nun aber die Gleichung z µ + tµ−1z µ−1 · · · = 0 irreducibel ist, so muß sie, weil sie vom µ − 1 ten
Grade ist, einzeln
α µ − 1 = 0, α − α µ = 0 . . . α µ−1 − α µ = 0
geben; was nicht sein kann.” (ibid., 72).
25 ABEL carried out his deductions in ignorance of RUFFINI’S work (see section 6.7).
26 “Wenn eine Gleichung algebraisch auflösbar ist, so kann man der Wurzel allezeits eins solche Form
geben, daß sich alle algebraische Functionen, aus welchen sie zusammengesetzt ist, durch rationale
Functionen der Wurzeln der gegebenen Gleichung ausdrücken lassen.” (ibid., 73).