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

6.9. Reception of ABEL’s work on the quintic 129
Figure 6.2: WILLIAM ROWAN HAMILTON (1805–1865)
nothing of KÜLP’S attitude toward the validity of ABEL’S result. KÜLP’S criticism fo-
cused on two individual parts of ABEL’S argument. The first question was concerned
with a misprint which occurred in ABEL’S proof of the CAUCHY-RUFFINI theorem.
Due to the relatively new character of the theory of permutations and their notation,
KÜLP apparently had trouble following ABEL’S argument and was halted by the mis-
print. ABEL’S notation was apparently also a problem for KÜLP; in his answer, ABEL
proved the claim that any 3-cycle could be decomposed as the product of two p-cycles
by writing out the substitutions in detail. I mention these objections in order to illus-
trate the difficulties, conceptual and technical, which nineteenth century mathemati-
cians had in understanding and accepting ABEL’S proof.
KÜLP’S other objection concerned ABEL’S descriptive classification of rational func-
tions of five quantities which have five values. Again, we do not have KÜLP’S formu-
lation but only ABEL’S reply which ABEL posted from Paris less than a year after his
paper had appeared in CRELLE’S Journal. The argument given in the letter differed
substantially from the published one. As I have discussed above (in section 6.6.1),
the original argument was, indeed, very hard to understand. If ABEL’S refined proof
communicated to KÜLP had made it into print, ABEL’S conclusion might have been
accepted at an earlier point.