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

160 Chapter 7. Particular classes of solvable equations
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✛ Solvable equations ✲
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Polynomial equations
Figure 7.3: Extending the class of solvable equations: Abelian equations
7.4 Enlarging the class of solvable equations
ABEL considered the positive result demonstrating the solubility of certain equations
as a counterpart to the insolubility of higher degree general equations. In the intro-
duction to the Mémoire sur une classe particulière, ABEL wrote:
“It is true that the algebraic equations are not generally solvable, but there is a
particular class of each degree for which the algebraic solution is possible.” 47
To this class of solvable equations belonged the equations of the form x n − 1 = 0
studied by GAUSS and the generalizations of these obtained by ABEL in the paper.
Only few other equations were explicitly known to be solvable, and ABEL’S result
can thus be seen to provide a demonstration that the total class of solvable equations
had a certain range. In the limitation-enlargement model suggested in section 6.8,
the situation can be described by figure 7.3 and much of ABEL’S research to describe
the precise extent of solubility can be interpreted in this context. In a letter to B. M.
HOLMBOE (1795–1850) written during his stay in Paris, ABEL described the problem
and his progress:
“I am currently working on the theory of equations, which is my favorite
theme, and have finally reached a point where I see a way to solve the following
general problem: To determine the form of all algebraic equations which can
be solved algebraically. I have found an infinitude of the fifth, sixth, seventh, etc.
degree which had never been smelled before.” 48
47 “Il est vrai que les équations algébriques ne sont pas résolubles généralement; mais il y en a une
classe particulière de tous les degrés dont la résolution algébrique est possible.” (N. H. Abel, 1829c,
131).
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