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

8.5. General resolution of the problem by E. GALOIS 185
Thus, GALOIS had characterized solvable irreducible equations of prime degree p
by the necessary and sufficient requirement that their GALOIS group contained nothing
but permutations corresponding to the linear congruences 53
i → ai + b (mod p) where p ∤ a. (8.10)
From this characterization of solubility, GALOIS deduced a second one which ABEL
had also hit upon (see section 8.4), when he demonstrated his eighth proposition:
“Theorem. For an equation of prime degree to be solvable by radicals it is necessary
and sufficient that any two of its roots being known, the others can be deduced
rationally from them.” 54
The character of GALOIS’ reasoning often left quite a lot to be desired. When LI-
OUVILLE eventually published GALOIS’ manuscripts, he accompanied them with an
evaluation of GALOIS’ clarity and rigour:
“Clarity is indeed an absolute necessity. [. . . ] Galois too often neglected this
precept.” 55
In making GALOIS’ new ideas available to the mathematical community and in
providing proofs and elaborations of obscure points, mathematicians of the second
half of the nineteenth century invested much effort in the theory of equations, per-
mutations, and groups. Although GALOIS had found out how the solubility of a
given equation could be determined by inspecting the decomposability of its asso-
ciated group into a tower of normal subgroups, a number of points were left open
for further research. To mathematicians around 1850, three problems were of primary
concern: GALOIS’ construction of the group of an equation was considered to be unrig-
orous, no characterization of the important solvable groups had been carried out, and a
certain arbitrariness of the order of decomposition also remained. These matters were
cleared, one by one, until the theory ultimately found its mature form in the abstract
field theoretic formulation of H. WEBER (1842–1913) and E. ARTIN (1898–1962). 56
8.5.2 Common inspiration and common problems
As mentioned earlier (p. 181), GALOIS and ABEL drew extensively on common sources.
The ideas of invariance under permutations of the roots, founded in LAGRANGE’S
work, 57 were important to both of them; and they both relied on the general theory
53 (Scholz, 1990, 385).
54 “Théorème. Pour qu’une équation de degré premier soit soluble par radicaux, il faut et il suffit que
deux quelconques des racines étant connues, les autres s’en déduisent rationnellement.” (Galois,
1831c, 69).
55 (Liouville quoted from Kiernan, 1971, 77).
56 (ibid.) and (Scholz, 1990, 392–398).
57 (Lagrange, 1770–1771)