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

8.5. General resolution of the problem by E. GALOIS 183
LOIS, he was instrumental in bringing about the structural approach to mathematics,
which came to dominate much of 20 th century mathematics. 43
GALOIS’ work was, as he himself somewhat laconically remarked, 44 founded in
the theory of permutations most of which he had taken over from CAUCHY. GALOIS
considered an equation of degree m
φ (x) = 0
having the roots x1, . . . , xm, and claimed that the group of the equation G — later called
the Galois group — could always be found, which had the following two properties:
1. that every function of the roots x1, . . . , xm which was (numerically) invariant un-
der the substitutions of G was rationally known, and conversely,
2. that every rational function of the roots x1, . . . , xm was invariant under the sub-
stitutions of G.
GALOIS took over the concept of rationally known from LAGRANGE but changed
the notion of invariant to stress numerical invariance instead of LAGRANGE’S formal in-
variance in order to deal with special (i.e. non-general) equations. However, GALOIS’
proof of the existence of the group of the equation suffered from the unclear character
of his concept of invariance. 45
Although the concepts of permutation and substitution underwent some uncom-
pleted changes in GALOIS’ manuscripts, he clearly perceived the multiplicative nature
of substitutions — understood as transitions from one arrangement (permutation) to
another — as well as the multiplicative closure of the GALOIS group.
“It is clear in the group of permutations under consideration, the arrangement
of letters is not important, but only the substitutions on the letters, by which we
move from one permutation to another. Thus, if in similar group one has the
substitutions S and T, one is also certain to have the substitution ST.” 46
The second component of GALOIS’ theory addressed the reduction of the group
of an equation by the adjunction of quantities to the set of rationally known quantities.
By adjoining to the rationally known quantities a single root of an irreducible aux-
iliary equation, GALOIS could decompose the group of the equation into a number,
p, of subgroups. These had the remarkable property that applying a substitution to
43 These aspects of GALOIS’ work have been studied by, for instance, (Wussing, 1969) and (Kiernan,
1971).
44 (Galois, 1830, 165).
45 (Kiernan, 1971, 80–81).
46 “Comme il s’agit toujours de questions où la disposition primitive des lettres n’influe en rien, dans
les groupes que nous considérons, on devra avoir les mêmes substitutions quelle que soit la permutation
d’où l’on sera parti. Donc si dans un pareil groupe on a les substitutions S et T, on est sûr
d’avoir la substitution ST.” (Galois, 1831c, 47). I have extended the translation found in (Kiernan,
1971, 80).