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

136 Chapter 6. Algebraic insolubility of the quintic
proof. ANDERSSEN found the proof to be simple, coherent, and not built upon calcu-
lations but on arguments and deductions; at the same time, and possibly for the same
reasons, he rated it as being difficult.
“However simple this proof is, first of all because a single idea serves throughout
as a decisive criterion, secondly because the truths by which the application
of the main idea is possible, are communicated not by artificial calculations but
by conclusions and deductions, it nevertheless (and even therefore) demands the
most thorough contemplation in order to be understood in its entire clarity. Therefore
it would not be a superfluous work to present the most important arguments
of this instructive yet difficult proof by examples and further elaborations in their
true spirit and full power of proof.” 108
ABEL’S classification of algebraic expressions according to orders and degrees was
reproduced in an overly simplified form, in which the concept of degree has com-
pletely vanished. When it came to the classification of functions with five quantities,
which HAMILTON had scrutinized, ANDERSSEN found it quite satisfactory:
“Both these two theorems [no function of five quantities can have two or
five different values under all possible interchanges of the quantities] have been
proved in Abel’s treatise with a clarity which cannot be improved.” 109
ANDERSSEN’S essay contained no criticism of parts of ABEL’S proof nor any origi-
nal modifications but only simple elaborations and some examples. However, its mere
existence is evidence that ABEL’S result was becoming known to the broader circle of
German mathematicians.
LEOPOLD KRONECKER. The introduction of ABEL’S work on the quintic equation
into German academic circles is due to LEOPOLD KRONECKER. Much of KRONECKER’S
work on algebra was inspired by ideas which he got reading ABEL and KRONECKER
completed and rigorized many parts of ABEL’S research. In KRONECKER’S elegant
proof of the insolubility of the general fifth (and higher) degree equation, ABEL’S
proof found its final form. KRONECKER presented his simplified version of ABEL’S
proof in a paper read to the Akademie der Wissenschaften. 110 There, he presented no
criticism of ABEL’S proof but simply put forward alternative deductions preferable to
ABEL’S on account of their simplicity and general nature. The validity of the result
108 “So einfach dieser Beweis ist, erstens weil ein einziger Gedanke durchgehends zum entscheidenden
Kriterium dient, zweitens weil diejenigen Wahrheiten, kraft deren die Anwendung des Hauptgedankens
möglich ist, nicht durch künstliche Rechnungen, sondern durch Urtheile und Schlüsse
vermittelt werden; so erheischt er dennoch, ja eben deswegen das gesammeltste Nachdenken, um in
seiner ganzen Klarheit begriffen zu werden. Es dürfte daher keine unnöthige Arbeit sein, die wichtigsten
Argumente dieses eben so lehrreichen als schwierigen Beweises durch Beispiele und weitere
Ausführung in ihrem wahren Sinne und in ihrer vollständigen Beweiskraft zur Anschauung zu
bringen.” (Anderssen, 1848, 3).
109 “Diese beiden Lehrsätze sind in Abel’s Abhandlung mit einer Klarheit bewiesen, welche durch
Nichts erhöht werden kann.” (ibid., 14).
110 (Kronecker, 1879).