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

8.2. Construction of the irreducible equation 169
1
µ
be satisfied by all values of y which are obtained by attributing to y1
1
µ
ωy1
, ω2 1
µ
y1
, . . . , ωn−1 1
µ
y
all the values
1 ”12 (see section 6.3.3). In 1826, it had been given no proof, but
in the notebook, ABEL provided the proof as an easy and elegant application of the
fundamental concepts and tools.
ABEL proceeded by establishing a central link between the irreducibility of φ (y, m) =
0 and that of ∏ φ (y, m) = 0.
Theorem 11 If the equation
is irreducible, then so is the equation
φ (y, m) = 0
φ1 (y, m) = ∏ φ (y, m) = 0. ✷
ABEL argued for this theorem by a reductio ad absurdum proof against which SYLOW
later raised well founded objections. ABEL assumed that φ1 was reducible and that
φ2 (y, m ′ ) was an irreducible 13 factor of ∏ φ (y, m) = 0. Under these assumptions,
φ2 and φ would have a common root since all the roots of ∏ φ were also roots of
φ. The assumed irreducibility of φ then enabled ABEL to conclude that because the
irreducible φ and φ2 had a root in common, φ would be a factor of φ2,
φ2
This in turn implied (by theorem 10)
φ2
� y, m ′ � = f (y, m) · φ (y, m) .
� y, m ′ � = f1
� ′′
y, m � · ∏ φ (y, m) . (8.4)
� �� �
=φ1(y,m) On the other hand, φ2 had been assumed to be an irreducible factor of φ1 implying
deg φ2 < deg φ1, which contradicted (8.4).
SYLOW’S objections concerned the properties of ∏ φ. Besides certain points, at
which ABEL left out assumptions of irreducibility, SYLOW noticed that ABEL tacitly
assumed that φ (y, m) did not have factors in which all the coefficients were rational
expressions in inner radicals and known quantities. If such factors were involved, the
equation ∏ φ (y, m) = 0 might turn out to be a power of an irreducible equation. 14
SYLOW repaired ABEL’S argument by refining his hierarchy of algebraic expressions.
12 “[. . . ] so ist klar, daß der gegebenen Gleichung durch alle die Werthe von y genug werden muß,
welche man findet, wenn man der Größe p 1 n alle die Werthe αp 1 n , α 2 p 1 n , . . . , α n−1 p 1 n beilegt.” (N. H.
Abel, 1826a, 72).
13 Actually, ABEL did not, presumably inadvertently, state the condition of irreducibility of φ2.
14 (Sylow in N. H. Abel, 1881, vol. 2, 332).