to Abel, his papers had been generating increasing excitement in the
mathematical community, and two days after his death a letter arrived
bearing an offer of an academic position in Berlin.
Évariste Galois
The third major player in the solution of the quintic also suffered simi-
larly from bad luck. Évariste Galois was born nine years later than Abel in
a suburb of Paris. The son of a mayor, he did not exhibit any exceptional
ability in school—but by age sixteen he realized that despite the judg-
ments of his teachers, he possessed considerable mathematical talents.
He applied to the École Polytechnique, a school which had been attended
by many celebrated mathematicians, but his mediocre performance in
school prevented him from being accepted. He wrote a paper and pre-
sented it to the academy at age seventeen—but Augustin-Louis Cauchy,
one of the leading mathematicians of the era, lost it. He submitted an-
other paper to the academy shortly thereafter—but Joseph Fourier, the
secretary of the academy, died soon after the receipt of the paper and it,
too, was lost. Jonathan Swift once remarked that one could recognize gen-
ius by the fact that the dunces would conspire against them; Galois seems
to have been particularly unfortunate in that geniuses conspired against
him, albeit inadvertently.
Frustrated by all this incompetence, Galois sought an outlet in the poli-
tics of the times, and joined the National Guard. An active revolutionary,
in 1831, he proposed a toast at a banquet that was viewed as a threat
against King Louis Philippe. This declaration was followed by a mistake
that was to prove fatal—he became involved with a young lady whose
other lover challenged Galois to a duel. Fearing the worst, Galois spent
the night before the duel jotting down his mathematical notes, entrust-
ing them to a friend who would endeavor to have them published. The
duel took place the next day, and Galois died from his wounds a day later.
He was barely twenty years old.
Although Abel was the first to show the insolvability of the quintic, Ga-
lois discovered a far more general approach to the problem that was to be
of great significance. Galois was the first to formalize the mathematical
concept of a group, which is one of the central ideas in modern algebra.
The connection between groups, polynomial equations, and fields is one
of the primary themes of the branch of mathematics known as Galois
theory. Galois theory not only explains why there is no general solution to
the quintic, it also explains precisely why polynomials of lower degree
94 How Math Explains the World