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

7.3. The concept of irreducibility at work 159
further comment. 43 By inserting into (7.15), ABEL found
f (x) =
P + Q · φ (x)
. (7.16)
N
Next, he let x denote a common root of φ and f and concluded that x would also be
a root of P = 0. However, if P were not identically zero, “this equation gives x as a
root of an equation of degree less than that of φx = 0; which is a contradiction of the
hypothesis. Therefore, P = 0 and it follows that f x = φx · Q
N .”44 Thus, it was obvious
that f would vanish whenever φ did and, therefore, that any root of φ (x) = 0 would
also be a root of f (x) = 0.
ABEL put this important theorem to use in the very first description of the equa-
tions treated in the Mémoire sur une classe particulière. If x ′ and x were two roots of the
irreducible equation φ (x) = 0 among which a rational dependency existed,
x ′ = θ (x) ,
then every iterated application of θ to x would also be a root of this equation. ABEL’S
demonstration followed directly from the theorem above. He argued that since it fol-
lowed from the hypothesis that the equations
φ (θ (x)) = 0 and φ (x) = 0
had a root, x, in common, theorem 7.3 stated that for any root, y, of φ (x) = 0, θ (y)
would also be a root of that equation. Once he had established this result, the argu-
ment of ABEL’S paper was on its way, and the complex of conclusions described above
could be obtained.
ABEL turned the concept of irreducibility of equations, which had existed as an
ad hoc tool before into a central foundation upon which a building of theorems could
be established. 45 The irreducibility in ABEL’S sense was defined as minimality of the
equation expressing the roots under the restriction that the coefficients must depend
rationally on the same quantities as the original equation. From this definition, gen-
eralizations were later made toward the general concept of domain of rationality. But
working with this definition — and the division algorithm of EUCLID (∼295 B.C.) —
ABEL demonstrated the important theorem 9 of divisibility, which in turn established
the basic property of the class of equations studied in the Mémoire sur une classe particulière.
46
43 It had been explicitly employed in GAUSS’ second proof of the Fundamental Theorem of Algebra (see
section 5.7).
44 “cette équation donnera x, comme racine d’une équation d’un degré moindre que celui de φx = 0;
ce qui est contre l’hypothèse; donc P = 0 et par suite f x = φx · Q
45
N .” (ibid., 133,footnote).
See also (L. Sylow, 1902, 23–24).
46 (N. H. Abel, 1829c).