- THEORY OF EQUATIONS 449
all concrete objects—permutations of the roots of equations. He developed Galois
theory to the extent of analyzing what happened to the group of an equation when,
in modern terms, a new element is adjoined to the base field. Galois could not be
so clear. He said, "When we agree thus to regard certain quantities as known, we
shall say that we are adjoining them to the equation being solved and that these
quantities are adjoined to the equation." He thought of the new element as a root
of an auxiliary polynomial (the minimal polynomial of the new element, in our
terms), since that is where he got the elements that he adjoined. Instead of saying
that the original group might be decomposed into the cosets of the group of the
new equation when all the roots of the auxiliary equation are adjoined, he said it
might split into ñ groups, each belonging to the equation. He noted that "these
groups have the remarkable property that one can pass from one to the other by
operating on all the permutations with the same letter substitution."
In a letter to a friend written the night before the duel in which he died, Galois
showed that he had gone still further into this subject, making the distinction
between proper and improper decompositions of the group of an equation, that is,
the distinction we now make between normal and nonnormal subgroups.
Galois theory. The ideas of Galois and his predecessors were developed further
by Laurent Wantzel (1814-1848) and Enrico Betti. In 1837 Wantzel used Galois'
ideas to prove that it is impossible to double the cube or trisect the angle using
ruler and compass; in 1845 he proved that it is impossible to solve all equations
in radicals. In 1852 Betti published a series of theorems elucidating the theory of
solvability by radicals. In this way, group theory proved to be the key not only
to the solvability of equations but to the full understanding of classical problems.
When Ferdinand Lindemann (1852-1939) proved in 1881 that ð is a transcendental
number, it followed that no ruler-and-compass quadrature of the circle was possible.
The proof that the general quintic equation of degree 5 was not solvable by
radicals naturally raised two questions: (1) How can the general quintic equation
of degree 5 be solved? (2) Which particular quintic equations can be solved by
radicals? These questions required some time to answer.
Solution of the general quintic by elliptic integrals. A partial answer to the first
question came from the young mathematician Ferdinand Eisenstein (1823-1852),
who showed in 1844 that the general quintic equation could be solved in terms of
a function \(\) that satisfies the special quintic equation(÷(ë))^5 + ÷(ë) = ë,
This function is in a sense an analog of root extraction, since the square root
function ö and the cube root function ø satisfy the equations(ö{\))^2 = A,{mf = ë.
Eisenstein's solution stands somewhat apart from the main line of development,
but in modern times it begins to look more reasonable. To solve all quadratic
equations in a field of characteristic 2, for example, it is necessary to assume, in
addition to the possibility of extracting a square root, that one has solutions to the
equation ÷^2 + ÷ + 1 = 0; these roots must be created by fiat. For a full discussion
of Eisenstein's paper, see the article of Patterson (1990).