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

7.4. Enlarging the class of solvable equations 161
Thus, ABEL’S two publications on the theory of equations which appeared during
his lifetime contributed a negative, limiting result of insolubility of the general higher
degree equations and a positive, enlarging result of the solubility of a certain class of
equations of all degrees. The program set out above in the letter to HOLMBOE was
pursued by ABEL from his time in Paris, and traces of it can be found in his note-
books. However, his correspondence also announced further and far-reaching results
for which no detailed studies or proofs have been recovered. The determination of
the exact extension of the concept of algebraic solubility was approached by ABEL
through a theory largely based on the same tools as his published works, but never
completed nor published in his lifetime. Therefore, the solution to this fundamental
problem is rightfully attributed to E. GALOIS (1811–1832). In the next chapter, ABEL’S
steps toward a general theory of solubility are analyzed against the background of his
other works and GALOIS’ contemporary ideas.
48 “Jeg arbeider nu paa Ligningernes Theorie, mit Yndlingsthema og er endelig kommen saa vidt at jeg
seer Udvei til at løse følgende alm: Problem. Determiner la forme de toutes les équation algébriques
qui peuvent être resolues algebriquement. Jeg har fundet en uendelig Mængde af 5te, 6te, 7de etc.
Grad som man ikke har lugtet indtil nu.” (Abel→Holmboe, Paris, 1826/10/24. N. H. Abel, 1902a,
44).