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

Chapter 4
The position and role of ABEL’s works
within the discipline of algebra
In the nineteenth century, the theory of equations acquired its status as a mathematical
discipline with its own set of problems, methods, and legitimizations. In the process,
N. H. ABEL (1802–1829) played an important role. His works on the algebraic in-
solubility of the general quintic equation and his penetrating studies of the so-called
Abelian equations belong to the first results established within this incipient discipline.
Although ABEL’S investigations raised new questions and answered some of them,
his methods and his approach was deeply rooted in the works of mathematicians be-
longing to the previous generations. In particular, ABEL drew upon the algebraic re-
searches of L. EULER (1707–1783). Therefore, in the following chapter 5, these works,
similar approaches taken by A.-T. VANDERMONDE (1735–1796), and the even more
influential works by J. L. LAGRANGE (1736–1813) and C. F. GAUSS (1777–1855) are
described and analyzed. In the ensuing chapters 6–8, ABEL’S algebraic researches are
described and their role and impact are analyzed. Focus in this part II will be on de-
scribing the change in asking and answering questions pertaining to mathematical ob-
jects; more precisely questions concerning the algebraic solubility of equations. Such
questions have been central to mathematical development since the Renaissance, but
starting in the second half of the eighteenth century, they gave rise to a new mathe-
matical theory. Once this theory-building has been described, the attention is directed
toward ABEL’S approach to algebraic questions. ABEL’S studies of the quintic equa-
tion provide an example of how a change in the process of asking questions led to
unexpected answers. Then, because of the similarity in methods and inspirations,
ABEL’S questions concerning the geometric division of the lemniscate are treated to
illustrate how an algebraic topic emerged within an apparently non-algebraic realm.
Finally, the quest — taken up by ABEL and slightly later by E. GALOIS (1811–1832) —
to completely describe solvable equations is outlined to provide a first illustration of
the new and more abstract kind of questions which C. G. J. JACOBI (1804–1851) saw
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