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

184 Chapter 8. A grand theory in spe
permutations in one of the subgroups gave the permutations of another subgroup. 47
When GALOIS adjoined the entire set of roots of the irreducible auxiliary equation, he
obtained an even more remarkable result:
“Theorem. If one adjoins to an equation all the roots of an auxiliary equation,
the groups in question in theorem II [i.e. the p subgroups mentioned above]
will furthermore have the property that the substitutions are the same in each
group.” 48
Of this important theorem GALOIS gave no proof, but hastily remarked “the proof
will be found.” 49 The contents of the theorem is GALOIS’ characterization of the defin-
ing property of what was be called normal subgroups, since GALOIS’ statement corresponds
to saying that all the conjugate classes of a subgroup U are identical. 50
The link between properties of the decomposition into normal subgroups of the
group of the equation and the algebraic solubility of the equation was provided in the
far-reaching fifth problem of the manuscript. Using modern concepts and terms, it
can be summarized as follows. Assuming that the equation under consideration had
the group G, and that p was the smallest prime divisor of the number of permutations
in G, GALOIS argued that the equation could be reduced to another equation having
a smaller group G ′ whenever a normal subgroup N existed in G with index p. Fur-
thermore, the link with algebraic solubility was provided when GALOIS stated that the
equation would be solvable in radicals precisely when its group could be decomposed
into the trivial group by iterated applications of the preceding principle. 51
GALOIS applied the general result on algebraic solubility in two ways to obtain
important characterizations of solubility of equations. First, he sought criteria for sol-
ubility of irreducible equations of prime degree and found the following:
“Thus, for an irreducible equation of prime degree to be solvable by radicals it
is necessary and sufficient that any function which is invariant under the substitutions
x k x ak+b
[a and b are integer constants] is rationally known.” 52
47 (Galois, 1831c, 55).
48 “Théorème. Si l’on adjoint à une équation toutes les racines d’une équation auxiliaire, les groupes
dont il est question dans le théorème II jouiront de plus de cette propriété que les substitutions sont
les mêmes dans chaque groupe.” (ibid., 57).
49 “On trouvera la démonstration.” (ibid., 57).
50 (Scholz, 1990, 384).
51 (ibid., 384–385).
52 “Ainsi, pour qu’une équation irréductible de degré premier soit soluble par radicaux, il faut et il
suffit que toute fonction invariable par les substitutions
x k x ak+b
soit rationnellement connue.” (Galois, 1831c, 69).