448 15. MODERN ALGEBRAwe associate with a group: the composition of permutations. However, it is clear
from his language that what makes a set of permutations a group is that all of them
have the same effect on certain rational functions of the roots. In particular, when
what we now call a group is decomposed into cosets over a subgroup, Galois refers
to the cosets as groups, since any two elements of a given coset have the same effect
on the rational functions. He says that a group, in this sense, may begin with any
permutation at all, since there is no need to specify any natural initial order of the
roots.
Besides the shortness of their lives, Abel and Galois had another thing in com-
mon: neglect of their achievements by the Paris Academy of Sciences. We shall see
some details of Abel's case in Chapter 17. As for Galois, he had been expelled from
the Ecole Normale because of his Republican activities and had been in prison. He
left a second paper on the subject among his effects, which was finally published
in 1846.^10 It had been written in January 1831, 17 months before his death, and
it contained the following plaintive preface:The attached paper is excerpted from a work that I had the honor
to present to the Academy a year ago. Since this work was not un-
derstood, and doubt was cast on the propositions that it contains,
I have had to settle for giving the general principles and only one
application of my theoric in systematic order. I beg the referees at
least to read these few pages with attention. [Picard, 1897, p. 33]The language and notation used by Galois are very close to those of Lagrange.
He considers an equation of degree ç and claims that there exists a function (polyno-
mial) ö(á, b,c,d,...) that takes on n! different values when the roots are permuted.
Such a polynomial, he says, can be ö(á,b,c,d,...) = Aa + Bb + Cc + Dd + • • •,
where A, B, C, D, and so on, are positive integers. He then fixes one root a and
forms a function of two variables= Ð (V-v(o,ft,c,d,...)),(the Galois resolvent), in which the product extends over all permutations that
leave a fixed. Since the function on the right is symmetric in b, c, d,..., all of
these variables can be replaced by suitable combinations of á and the coefficients
pi,...,pn (see Problem 15.4). The equation f(ip(a,b,c,d,...),x) = 0 then has
the solution ÷ = a, but has no other roots in common with the equation p(x) =
- Finding the greatest common divisor of these two polynomials then makes it
possible to express á as a rational function of ø(á,b,c,d,...). Galois cited one of
Abel's memoirs (on elliptic functions) as having stated this theorem without proof.
The main theorem of the memoir was the following: For any equation, there
is a group of permutations of the roots such that every function of the roots that
is invariant under the group can be expressed rationally in terms of the coefficients
of the equation, and conversely, every such function is invariant under the group.
We would nowadays say that the elements of this group generate automorphisms
of the splitting field of the equation that leaves the field of coefficients invariant.
As his formulation shows, Galois had only the skeleton of that result. He called
the group of permutations in question the group of the equation. His groups are
(^10) Abel's great work on integrals of algebraic functions, submitted in 1827, was finally published,
at the insistence of Jacobi, in 1841.