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

8.5. General resolution of the problem by E. GALOIS 181
The major parts of ABEL’S research on equations which can be rendered intelligible
have been presented above. Nevertheless, ABEL’S notebooks are filled with notes and
scribbles for additional research which he never translated into a finished form suit-
able for presentation. During the few remaining years of his life, ABEL became pre-
occupied with other mathematical topics. Thus, we can only wonder what he might
have achieved, had he returned to the theory of solubility per se.
8.5 General resolution of the problem by E. GALOIS
ABEL’S attempt at a general theory of the algebraic solubility of equations was not
published until the first edition of the Œuvres 1839. Hence, it is most likely that GA-
LOIS was unaware of ABEL’S general research when he wrote down his theory in the
early 1830s. GALOIS knew the published works of LAGRANGE and A.-L. CAUCHY
(1789–1857), and he had probably read ABEL’S two publications on the theory of equa-
tions — the impossibility proof of 1826 and the Mémoire sur une classe particulière pub-
lished 1829 33 — as well as ABEL’S more widely known works on the theory of elliptic
functions, the Recherches sur les fonctions elliptiques 34 and the Précis d’une théorie des
fonctions elliptiques 35 . 36 GALOIS “vehemently denied” 37 dependence on ABEL as can
be seen from the fragmentary Note sur Abel, 38 but undeniably they share many of their
inspirations. In section 8.5.1, I briefly describe GALOIS’ unified theory before I com-
ment upon the common inspiration and central problems shared in the works of ABEL
and GALOIS (section 8.5.2).
The turbulent life of EVARISTE GALOIS as well as the interplay between his life
and the fate of his mathematics have been studied intensively. 39 GALOIS’ theory of
algebraic solubility was not made public to the mathematical community except for a
small group of members of the Institut de France until J. LIOUVILLE (1809–1882) pub-
lished selections from GALOIS’ mathematical manuscripts in the Journal de mathéma-
tiques pures et appliquées in 1846. 40 Subsequently, many mathematicians in the sec-
ond half of the nineteenth century invested great efforts in incorporating GALOIS’
at times fragmentary and non-rigorous mathematics into the new standards of clar-
ity and rigour. The process made mathematicians like KRONECKER return to ABEL’S
works and manuscripts (see section 6.9.2), but was largely an enterprise of digesting
GALOIS’ work. Therefore, the reception of GALOIS’ theory is not the primary concern
33 (N. H. Abel, 1826a; N. H. Abel, 1829c)
34 (N. H. Abel, 1827b; N. H. Abel, 1828b)
35 (N. H. Abel, 1829d)
36 (Wussing, 1969, 75).
37 (Kiernan, 1971, 90).
38 (Galois, 1831b).
39 For instance (Wussing, 1975), (Rothman, 1982), or (Toti Rigatelli, 1996).
40 (Lützen, 1990, 559–580).