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

6.1. The first break with tradition 99
In this chapter, I give a presentation of ABEL’S proof using the tools and methods
available to him. As described in the introduction, 9 this approach allows me to place
ABEL’S proof in a historical context within mathematics. For expositions of ABEL’S
proof involving the modern concepts introduced in Galois theory, see for instance (R.
Ayoub, 1982; M. I. Rosen, 1995; Skau, 1990).
6.1 The first break with tradition
In the opening paragraph of the paper in CRELLE’S Journal für die reine und angewandte
Mathematik, ABEL described the approach he had taken. In order to answer the ques-
tion of solubility of equations, he proposed to investigate the forms of all algebraic
expressions in order to determine if they could “solve” the equation. Although ABEL
throughout spoke of algebraic functions, I use the term algebraic expressions to avoid any
confusion with the modern concept of a function as a mapping between sets. The
algebraic expressions which ABEL considered were algebraic combinations of the co-
efficients of the given equation, and thus his approach was in line with the one taken
earlier by A.-T. VANDERMONDE (1735–1796) (see section 5.1). 10
“As is known, the algebraic equations up to the fourth degree can be solved
in general. Equations of higher degrees, however, only in particular cases, and if I
am not mistaken, the question:
Is it possible to solve equations of higher than the fourth degree in general?
has not yet been answered in a satisfactory manner. The present treatise is concerned
with this question.
To solve an equation algebraically is but to express its roots by algebraic functions
of its coefficients. Therefore, one must first consider the general form of algebraic
functions and subsequently investigate whether it is possible that the given
equation can be satisfied by inserting the expression of an algebraic function in
place of the unknown quantity.” 11
In the quote, ABEL also introduced an important notion of satisfiability. An equation
was said to be satisfied by an algebraic expression if the expression was a root of the
equation. Consequently, an equation was said to be satisfiable if an algebraic expres-
sion existed which satisfied it. This differed from the notion of algebraic solubility
which required that all the roots of the equation could be expressed algebraically.
9 See section 1.4.
10 (Kiernan, 1971, 67).
11 “Bekanntlich kann man algebraische Gleichungen bis zum vierten Grade allgemein auflösen, Gleichungen
von höhern Graden aber nur in einzelnen Fällen, und irre ich nicht, so ist die Frage:
Ist es möglich, Gleichungen von höhern als dem vierten Grade allgemein aufzulösen?
noch nicht befriedigend beantwortet worden. Der gegenwärtige Aufsatz hat diese Frage zum Gegenstande.
Eine Gleichung algebraisch auflösen heißt nichts anders, als ihre Wurzeln durch eine algebraische
Function der Coefficienten ausdrücken. Man muß also erst die allgemeine Form algebraischer Functionen
betrachten und alsdann untersuchen, ob es möglich sei, der gegebenen Gleichung auf die
Weise genug zu thun, daß man den Ausdruck einer algebraischen Function statt der unbekannten
Größe setzt.” (N. H. Abel, 1826a, 65).