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

Chapter 8
A grand theory in spe: algebraic
solubility
In his correspondence with A. L. CRELLE (1780–1855) and B. M. HOLMBOE (1795–
1850), N. H. ABEL (1802–1829) announced numerous results in the theory of equa-
tions beyond the impossibility of solving the quintic and the study of Abelian equa-
tions. Some concerned the form of solutions to algebraically solvable equations of the
fifth degree, 1 others dealt with solubility results for broader classes of equations, 2 and
yet others testify to ABEL’S general progress in his program of determining the form
of solvable equations. 3 The information provided in the letters is complemented by
a notebook entry dating from 1828 which has been included in both editions of the
Œuvres under the title Sur la résolution algébrique des équations. 4 The entry begins as a
manuscript almost ready for press, but after some introductory remarks, a few theo-
rems and some deductions it turns from its initial thoroughness and clarity to nothing
but bare calculations. Nevertheless, when considered together, these sources give an
impression of the methods and extent of the general theory of algebraic solubility
which ABEL set out to develop in the last years of his life.
8.1 Inverting the approach once again
The notebook manuscript dealt with the general form of algebraically solvable equa-
tions. In one of the two lengthy introductions which ABEL wrote for this work the
problem was clearly set out:
“Given an equation of any given degree, to determine whether or not it could
be satisfied algebraically.” 5
1 (Abel→Crelle, Freyberg, 1826/03/14. N. H. Abel, 1902a, 21–22).
2 (Abel→Crelle, Christiania, 1828/08/18. ibid., 72–73).
3 (Abel→Holmboe, Paris, 1826/10/24. ibid., 44–45).
4 (N. H. Abel, [1828] 1839).
5 “Une équation d’un degré quelconque étant proposée, reconnaître si elle pourra être satisfaite algébriquement,
ou non.” (N. H. Abel, 1881, vol. 2, 330).
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