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

350 Chapter 19. The Paris memoir
An important lemma
Lemma 4 Let χ (y) = 0 be an irreducible equation of degree n and let θ (y) be an equation
of degree n − 1. Then y can be expressed rationally in χ, θ. ✷
PROOF (PROOF OF LEMMA 4) By the Euclidean algorithm, there exist polynomials q, r
such that
χ = qθ + r
where deg r < deg θ. Since χ is irreducible and deg q = deg χ − deg θ = 1 > 0,
deg r > 0. Thus, there exist numbers s, t such that
Consequently,
y =
q (y) = sy + t.
q (y) − t
s
=
χ(y)−r
θ(y)
− t
,
s
and y has been expressed rationally in χ, θ. �
Box 9: An important lemma
which apparently was innovative with him. As is evident from even this exam-
ple, both the index over which the summation is to be performed and the upper
summation limit are implicit in the shorthand version.
2. For a rational function Fx, ABEL let ΠFx denote the coefficient of 1 x in the series
expansion of Fx in decreasing powers of x. Designating the ‘same’ object as the
residue which CAUCHY studied from an emerging perspective of his calculus of
residues, ABEL’S Π corresponds to CAUCHY’S E.
3. ABEL also introduced the operation h on algebraic functions which represented
a general degree of algebraic functions.
4. In the ultimate example of hyperelliptic integrals, ABEL introduced the notation
EA and εA for A any real number to denote the integer and remaining part, A =
EA + εA (EA ∈ Z and 0 ≤ εA < 1). It is worth remarking that ABEL did not — in
the Paris memoir considered as a whole — apply this notation consistently. Until
the ultimate section, he preferred the verbal formulation ‘the greatest integer
contained in the number’.
All these notational innovations enabled ABEL to comprehend, master, and manip-
ulate objects in a precise way which had hitherto been difficult to obtain.