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

Chapter 7
Particular classes of equations:
enlarging the class of solvable
equations
If N. H. ABEL’S (1802–1829) proof of the impossibility of solving the general quin-
tic algebraically was hampered by its brevity and obscure arguments, his only other
published work on the theory of equations was more mature, beautifully lucid, and
rigorous. In the Mémoire sur une classe particulière d’équations résolubles algébriquement,
written in 1828 and published the following year, ABEL abandoned one of the central
pillars of the impossibility proof — the theory of permutations — and provided a direct
and affirmative proof of the algebraic solubility of a particular class of equations. 1
Focusing instead on the other pillar — the concepts of divisibility, irreducibility, and
the Euclidean algorithm — this work illuminates central ideas in ABEL’S reasoning
which permeate his entire work on the theory of equations.
The 1829-paper has become a classic of mathematics for its proof that the class of
equations, now called Abelian and defined by certain properties of the roots, are always
algebraically solvable. When contrasted with the contents of the impossibility proof,
this result highlights a feature of the new ways of asking questions — the mechanisms
of limiting and enlarging class of objects which in the nineteenth century provided the
background for a new, concept based approach to mathematics. However, the paper
contains more information than just this main result; in this chapter I describe some of
the connections between this work and other parts of ABEL’S research as well as some
of the very central concepts which ABEL put to use in it.
1 (N. H. Abel, 1829c).
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