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

6.10. Summary 139
of ABEL’S argument but superior in rigour. The classification of algebraic expressions
was also a concern of some later 19 th century mathematicians, until it was settled by
KÖNIGSBERGER and KRONECKER.
The fact that global criticism of ABEL’S impossibility proof was limited can be taken
as a sign that the mathematical community soon came to realize the overall validity of
the result. The change of attitude toward the problem, which had been facilitated by
the statements of experts such as LAGRANGE and GAUSS (section 5.4) and the proofs
of RUFFINI, which at least were known in some circles in Paris, had been a prerequisite
for the quick acceptance. However, local criticism was still conducted in an effort to
make ABEL’S proof clearer and more powerful. Central lemmata, on which doubt
could be cast, were reexamined and new proofs were given.
6.10 Summary
As described, ABEL’S proof of the insolubility of the general quintic was a curious
combination of general theorems and investigations of particular cases. Partly be-
cause of the counter intuitive nature of the result and partly because of legitimate
local objections to ABEL’S argument, the result was subsequently scrutinized. Inter-
preted in terms of delineation of concepts, the algebraic insolubility of the general
quintic distinguished the concepts of polynomial equations and algebraically solvable
equations.