- THEORY OF EQUATIONS 447
of mathematics Leonard Eugene Dickson as "a very complicated reconstruction of
Abel's proof." Hamilton regarded the problem of the solvability of the quintic as
still open. He wrote:
[T]he opinions of mathematicians appear to be not yet entirely
agreed respecting the possibility or impossibility of expressing a
root as a function of the coefficients by any finite combination of
radicals and rational functions.The verdict of history has been that Abel's proof, suitably worded, is correct.
Ruffini also had a sound method (see Ayoub, 1980), but needed to make certain
subtle distinctions that were noticed only after the problem was better understood.
By the end of the nineteenth century, Klein (see 1884) referred to "the proofs of
Ruffini and Abel, by which it is established that a solution of the general equation
of the fifth degree by extracting a finite number of roots is impossible."
Besides his impossibility proof, Abel made positive contributions to the solution
of equations. He generalized the work of Gauss on the cyclotomic (circle-splitting)
equation xn +÷ç_1 +· • ---x + l = 0, which had led Gauss to the construction of the
regular 17-sided polygon. Abel showed that if every root of an equation could be
generated by applying a given rational function successively to a single (primitive)
root, the equation could be solved by radicals. Any two permutations that leave
this function invariant necessarily commute with each other. As a result, nowadays
any group whose elements commute is called an Abelian group.
1.9. Galois. More light was shed on the solution of equations by the work of
Abel's contemporary Evariste Galois (1811-1832), a volatile young man who did
not live to become even mature. As is well known, he died at the age of 20 in a
duel fought with one of his fellow Republicans.^9
The neatly systematized concepts of group, ring, and field that now make
modern algebra the beautiful subject that it is grew out of the work of Abel and
Galois, but neither of these two short-lived geniuses had a full picture of any of
them. The absence of the notion of a field seems to be the most noticeable lacuna
in the theorems they were proving. Where we now talk easily about algebraic and
transcendental field extensions and regard the general equation of degree ç over a
field F as ÷" + áé÷ç_1 + · · • + an-ix + o„ = 0, where a, is transcendental over
F, Galois had to explain that the concept of a rational function was relative to
what was given. For an equation with numerical coefficients, a rational function
was simply a quotient of two polynomials with numerical coefficients, while if the
equation had letters as coefficients, a rational function meant a quotient of two
polynomials whose coefficients were rational functions of the coefficients of the
equation. Even the concept of a group, which is associated with Galois, is not stated
formally in any of his work. He does use the word group frequently in referring to
a set of permutations of the roots of an equation, and he uses the properties that(^9) The word Republican (republicain) is being used in its French sense, of course, not the American
sense. It is approximately the opposite of royaliste. There are murky details about the duel, but
it appears that the gun Galois used was not loaded, probably because he did not wish to kill a
comrade-in-arms. It is also possible that the combatants had jointly decided to let fate determine
the outcome and each picked up a weapon not knowing which of the two guns was loaded. The
cause of the duel is also not entirely clear. The notes that Galois left behind seem to imply that
he felt it necessary to warn his friends about what he considered to be the wiles of a certain young
woman by whom he felt betrayed, and they felt obliged to defend her honor against his remarks.