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

4.1. Outline of ABEL’s results and their structural position 51
In the first part of the nineteenth century, the century-long search for algebraic
solution formulae was brought to a negative conclusion: no such formula could be
found. To many mathematicians of the late eighteenth century such a conclusion had
been counter-intuitive, but owing to the work and utterings of men like E. WARING
(∼1736–1798), 5 LAGRANGE, and GAUSS the situation was different in the 1820s.
ABEL’S proof was also met with criticism and scrutiny. By and large, though, the
criticism was confined to local parts of the proof. The global statement — that the gen-
eral quintic was unsolvable by radicals — was soon widely accepted.
Abelian equations. In his only other publication on the theory of equations, Mémoire
sur une classe particulière d’équations résolubles algébriquement 1829, ABEL took a different
approach. The paper was inspired by ABEL’S own research on the division problem
for elliptic functions and GAUSS’ Disquisitiones arithmeticae. In it, ABEL demonstrated
a positive result that an entire class of equations — characterized by relations between
their roots — were algebraically solvable.
For his 1829 approach, ABEL seamlessly abandoned the permutation theoretic pil-
lar of the insolubility-proof. Instead, he introduced the new concept of irreducibility
and — with the aid of the Euclidean division algorithm — proved a fundamental the-
orem concerning irreducible equations.
The equations which ABEL studied in 1829 were characterized by having rational
relations between their roots. 6 Using the concept of irreducibility, ABEL demonstrated
that such irreducible equations of composite degree, m × n, could be reduced to equa-
tions of degrees m and n in such a way that only one of these might not be solvable
by radicals. Furthermore, he proved that if all the roots of an equation could be writ-
ten as iterated applications of a rational function to one root, 7 the equation would be
algebraically solvable.
The most celebrated result contained in ABEL’S Mémoire sur une classe particulière
was the algebraic solubility of a class of equations later named Abelian by L. KRO-
NECKER (1823–1891). These equations were characterized by the following two prop-
erties: (1) all their roots could be expressed rationally in one root, and (2) these ratio-
nal expressions were “commuting” in the sense that if θ i (x) and θ j (x) were two roots
given by rational expressions in the root x, then
θ iθ j (x) = θ jθ i (x) .
By reducing the solution of such an equation to the theory he had just developed,
ABEL demonstrated that a chain of similar equations of decreasing degrees could be
constructed. Thereby, he proved the algebraic solubility of Abelian equations.
5 1734 is a more qualified guess for Waring’s year of birth than (Scott, 1976) giving “around 1736”.
See (Waring, 1991, xvi).
6 (N. H. Abel, 1829c).
7 I.e. an equation in which the roots are x, θ (x) , θ (θ (x)) , . . . , θ n (x) for some rational function θ.