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

18.1. Transformation theory 337
Summary: an algebraic proof. As described, ABEL’S deduction consisted of five
steps: First, ABEL set up his notation and definitions and introduced important results
from the Recherches. Second, ABEL found that if λ (θ (x)) is a root, i.e. if y =
ψ (λ (θ (x))), then any other root has the form λ (θ (x) + α). Next, the constant α could
be determined and a general representation of the roots can be given — possibly in-
volving multiple “orbits” corresponding to α1, α2, . . . . Fourth, the relation between y
and x could be spelled out. Eventually, it could be necessary to consider a number of
cases in order to describe these relations and deduce formulae of particular interest.
A few broader points should also be observed. First, ABEL’S proof made central
use of the properties of elliptic functions which had been deduced in the Recherches.
In particular, the solution of the equation λ (x) = λ (y) — which originated from the
double periodicity of the function λ — became very instrumental in the present con-
text just as it had been in the solution of the division problem (see section 16.3). Sec-
ond, the approach which ABEL took may well be described as an algebraic one; it relied
on algebraic tools such as specific knowledge of the roots of certain polynomial equa-
tions, division of polynomials, and considerations of the rationality of certain func-
tions. These are tools which were also present in ABEL’S purely algebraic researches
on solubility (see part II). However, ABEL also adopted another approach to the same
question.
Counting the possible numbers of transformations. In another paper — this time
published in CRELLE’S Journal and motivated by another of JACOBI’S papers — , 14
ABEL gave the theory of transformation a slightly different turn. He continued the
path laid out in the Astronomische Nachrichten and made frequent references to the
paper described above, but in the Journal, ABEL wanted to count and enumerate the
different transformations. ABEL considered a rational transformation of x into y of
a certain prime degree 2n + 1 and found — by employing algebraic tools similar to
those described above — that 12 (2n + 2) different transformations corresponding to
12 (n + 1) different values of the transformed modulus were generally possible. ABEL
remarked that for certain particular values of the modulus c, the number of transfor-
mations might degenerate. This notion of arguments carried out “in general” will be
discussed further in section 19.3 and chapter 21.
ABEL’S determination of the number of transformations spurred a reaction from
LEGENDRE who believed that it was at odds with JACOBI’S determination of the de-
gree of the so-called modular equation. JACOBI had claimed that for a transformation of
prime degree 2n + 1, 2n + 2 values of the transformed modulus were possible. Thus,
ABEL’S value was six times JACOBI’S number of transformations. However, as ABEL
argued in a letter to LEGENDRE, JACOBI had indeed solved an equation with 2n + 2
different roots but each root of this equation could also produce five other values for
14 (N. H. Abel, 1828e); JACOBI’S paper is (C. G. J. Jacobi, 1828).