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Methods
for solving quadratic equation were known to many ancient people,
including the Babylonian, Chinese and Hindu civilizations. Whether
or not they had the quadratic formula, they had the knowledge of
the main principle of completion of the square.
The history of the solution to the
cubic equation starts with Scipio del Ferro
(1465 -- 1526) who found a general formula for their solution.
His work was never published and was communicated only to a few friends. Cardano (1501 -- 1576) also discovered a formula for the
cubic
equation.
But for 350 years, no further progress was made. But
it was Galois,
pessimistic as he was, who showed that there is no algebraic
formula for solving polynomial equations of degree five or more.
Evariste Galois, a French
mathematician, was born in 1811. There is no sign of any
mathematical ability in any of Galois' family. He was taught by
his mother till he was 12 and then he enrolled at the Lycée of
Louis-le-Grand as a boarder in the 4th class. During 1824-25 his
school record was good and he received several prizes. However
in 1826 Galois was asked to repeat the year because his work in
rhetoric was not up to the required standard. And this weakness
would turn out to be a major drawback in his life.
In February 1827, he enrolled in
his first mathematics class, the class of M. Vernier. He quickly
became absorbed in mathematics and his director of studies
suggested that “it would be best for him if his parents would
allow him to study nothing but mathematics, and that he was
wasting his time there and did nothing but tormented his
teachers and overwhelmed himself with punishments”. M. Vernier
reported, “Intelligence, marked progress but not enough depth”.
Louis Richard reported, “This student works only in the highest
realms of mathematics”.
Galois
took the examination of the Ecole Polytechnique twice in 1828
and 1829, but failed both times and therefore resigned himself
to enter the Ecole Normale, which was an annex to Louis-le-Grand.
To do so he had to take his baccalaureate examinations, something
he could have avoided by entering the Ecole Polytechnique. He
passed, receiving his degree on 29 December 1829. His examiner
in mathematics reported, “This pupil is sometimes obscure in
expressing his ideas, but he is intelligent and shows a remarkable
spirit of research”.
In
April 1829, Galois had his first mathematics paper published on
continued fractions in the Annales de mathématiques.
On 25 May and 1 June, in 1829, he submitted articles on the algebraic
solution of equations to the Académie des Sciences. Cauchy was
appointed as referee of Galois' paper. Galois
sent Cauchy further work on the theory of equations, but then
learned from Bulletin de Férussac of a posthumous article
by Abel, which overlapped with a part of his work. Galois then
took Cauchy's advice and submitted a new article “On the condition
that an equation be soluble by radicals” in February 1830.
The paper was sent to Fourier, the secretary of the Academy, to
be considered for the Grand Prize in mathematics. Fourier died
in April 1830 and Galois' paper was never subsequently found and
hence never considered for the prize. Galois was invited by Poisson
to submit a third version of his memoir on the equation to the Academy
and he did so on 17 January, 1830.
14 July, 1830 was Bastille Day and Galois was arrested for the silly
crime of illegally wearing the uniform of the Artillery of the
National Guard. He was also carrying a loaded rifle, several pistols
and a dagger. Galois was sent to Sainte-Pélagie prison. While
in prison he received a rejection of his memoir. Poisson had reported
that: His argument is neither sufficiently clear nor sufficiently
developed to allow us to judge its rigor.
In
March 1832, Galois was transferred to the pension Sieur Faultrier.
There he apparently fell in love with Stephanie-Felice du Motel,
the daughter of the resident physician. After he was released
on 29 April 1832, Galois exchanged letters with Stephanie, and
it is clear that she tried to distance herself from the affair.
On May 30, Galois fought a duel with Perscheux d'Herbinville.
The
reason for the duel is not clear but certainly linked with
Stephanie. Galois was wounded in the duel and died in Cochin hospital,
Paris on 31 May 1832. His funeral was held on 2 June 1832.
Galois'
brother and his friend Chevalier copied his mathematical papers
and sent them to Gauss, Jacobi and others. It had been Galois'
wish that Jacobi and Gauss should give their opinions on his work
but no record exists of any comment these men made. However the
papers reached Liouville. In September 1843, Liouville announced to
the Academy that he had found in Galois' papers a concise solution,
“...as correct as it is deep of this lovely problem: Given
an irreducible equation of prime degree, decide whether or not
it is soluble by radicals”. In 1846, Liouville published these papers
of Galois in his journal called Journal de Mathématiques
Pures et Appliquées (also known as Journal de Liouvillin).
About
the author: This historical anecdote was brought to you by Jai
Paul, a graduate student in Mechanical Engineering at University
of South Florida - August 2002
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