Évariste Galois ( ) (October
25, 1811 – May 31, 1832) was a French mathematician born in Bourg-la-Reine. While still in his
teens, he was able to determine a
necessary and sufficient
condition for a
polynomial to be
solvable by
radicals, thereby solving a
long-standing problem. His work laid the foundations for
Galois theory, a major branch of
abstract algebra, and the subfield of
Galois connections. He was the
first to use the word "
group" (
) as a technical term in mathematics to represent a
group of
permutations. A
radical Republican during the
monarchy of
Louis Philippe
in France, he died from wounds suffered in a
duel under shadowy circumstances at the age of
twenty.
Life
Early life
Galois was born on October 25, 1811, to Nicolas-Gabriel Galois and
Adélaïde-Marie (born Demante). His father was a
Republican and was head of
Bourg-la-Reine's
liberal party, and
became mayor of the village after
Louis XVIII returned to the throne in
1814. His mother, the daughter of a
jurist,
was a fluent reader of
Latin and
classical literature and she was for
the first twelve years of her son's life responsible for his
education. At the age of 10, Galois was offered a place at the
college of Reims, but his mother
preferred to keep him at home.
In October 1823, he entered the Lycée
Louis-le-Grand, and despite some turmoil in the school at the
beginning of the term (where about a hundred students were
expelled), Galois managed to perform well for the first two years,
obtaining the first prize in Latin. He soon became bored
with his studies, and it was at this time, at the age of 14, that
he began to take a serious interest in mathematics. He found a copy
of
Adrien Marie Legendre's
Éléments
de Géométrie, which it is said that he read "like a novel"
and mastered at the first reading. At the age of 15, he was reading
the original papers of
Joseph
Louis Lagrange and
Niels Henrik
Abel, work intended for professional mathematicians, and yet
his classwork remained uninspired, and his teachers accused him of
affecting ambition and originality in a negative
way.
Budding mathematician
In 1828,
he attempted the entrance exam to École
Polytechnique, without the usual preparation in mathematics, and
failed for lack of explanations on the oral examination.
In that
same year, he entered the École préparatoire, a far inferior institution for mathematical
studies at that time, where he found some professors sympathetic to
him. In the following year, Galois' first paper, on
continued fractions was published, and
while it was competent it held no suggestion of genius.
Nevertheless, it was at around the same time that he began making
fundamental discoveries in the theory of
polynomial equations, and he submitted
two papers on this topic to the
Academy of Sciences.
Augustin Louis Cauchy refereed these
papers, but refused to accept them for publication for reasons that
still remain unclear. In spite of many claims to the contrary, it
appears that Cauchy had recognized the importance of Galois' work,
and that he merely suggested combining the two papers into one in
order to enter it in the competition for the Academy's Grand Prize
in Mathematics. Cauchy, a highly eminent mathematician of the time
considered Galois' work to be a likely winner (see below).On July
28, 1829, Galois' father
committed suicide
after a bitter political dispute with the village priest. A couple
of days later, Galois took his second, and final attempt at
entering the Polytechnique, and failed yet again. It is undisputed
that Galois was more than qualified; however, accounts differ on
why he failed. The legend holds that he thought the exercise
proposed to him by the examiner to be of no interest, and, in
exasperation, he threw the rag used to clean up chalk marks on the
blackboard at the examiner's head. More plausible accounts state
that Galois made too many logical leaps and baffled the incompetent
examiner, evoking irascible rage in Galois. The recent death of his
father may have also influenced his behavior.
Having
been denied admission to the Polytechnique, Galois took the
Baccalaureate examinations in order to enter the Ecole Normale. He passed, receiving his degree on December
29, 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."
His memoir on equation theory would be submitted several times but
was never published in his lifetime, due to various events. As
previously mentioned, his first attempt was refused by Cauchy, but
he tried again in February 1830 after following Cauchy's
suggestions and submitted it to the Academy's secretary
Fourier, to be considered for the Grand Prix
of the Academy. Unfortunately, Fourier died soon after, and the
memoir was lost. The prize would be awarded that year to Abel
posthumously and also to
Jacobi. Despite the lost memoir,
Galois published three papers that year, two of which laid the
foundations for
Galois theory, and the
third, an important one on
number
theory, where the concept of a
finite
field is first articulated.
Political firebrand
Galois lived during a time of political turmoil in France.
Charles X had succeeded Louis XVIII in
1824, but in 1827 his party suffered a major electoral setback and
by 1830 the opposition liberal party became the majority. Charles,
faced with abdication, staged a coup d'état, and issued his
notorious
July Ordinances, touching
off the
July Revolution which ended
with
Louis-Philippe
becoming king. While their counterparts at Polytechnique were
making history in the streets during the
les Trois
Glorieuses, Galois and all the other students at the École
Normale were locked in by the school's director. Galois was
incensed and he wrote a blistering letter criticizing the director
which he submitted to the
Gazette des Écoles, signing the
letter with his full name. Despite the fact that the
Gazette's editor redacted the signature for publication,
Galois was, predictably, expelled for it.
Even before his expulsion from Normale was to take effect on
January 4, 1831, Galois joined the staunchly Republican artillery
unit of the
National Guard.
These and other political affiliations continually distracted him
from mathematical work. Due to controversy surrounding the unit,
soon after Galois became a member, on December 31, 1830, the
artillery of the National Guard was disbanded out of fear that they
might destabilize the government. At around the same time, nineteen
officers of Galois' former unit were arrested and charged with
conspiracy to overthrow the government.
In April, all nineteen officers were acquitted of all charges, and
on May 9, 1831, a banquet was celebrated in their honor, with many
illustrious personalities, such as
Alexandre Dumas present. The
proceedings became more riotous, and Galois proposed
a toast to King Louis-Philippe with a dagger
above his cup, which was interpreted as a threat against the king's
life. He was arrested the following day, but was later acquitted on
June 15.
On the following
Bastille Day, Galois
was at the head of a protest, wearing the uniform of the disbanded
artillery, and came heavily armed with several pistols, a rifle,
and a dagger. For this, he was again arrested, this time sentenced
to six months in prison for illegally wearing a uniform. He was
released on April 29, 1832. During his imprisonment, he continued
developing his mathematical ideas.
Final days
Galois returned to mathematics after his expulsion from Normale,
although he was constantly distracted in this by his political
activities. After his expulsion from Normale was official in
January 1831, he attempted to start a private class in advanced
algebra which did manage to attract a fair bit of interest, but
this waned as it seemed that his
political
activism had priority.
Simeon
Poisson asked him to submit his work on the
theory of equations, which he submitted
on
January 17. Around
July 4, Poisson declared Galois' work
"incomprehensible", declaring that "[Galois'] argument is neither
sufficiently clear nor sufficiently developed to allow us to judge
its rigor"; however, the rejection report ends on an encouraging
note: "We would then suggest that the author should publish the
whole of his work in order to form a definitive opinion." While
Poisson's rejection report was made before Galois' Bastille Day
arrest, it took some time for it to reach Galois, which it finally
did in October that year, while he was imprisoned. It is
unsurprising, in the light of his character and situation at the
time, that Galois reacted violently to the
rejection letter, and he decided to forget
about having the Academy publish his work, and instead publish his
papers privately through his friend Auguste Chevalier. Apparently,
however, Galois did not ignore Poisson's advice and began
collecting all his mathematical manuscripts while he was still in
prison, and continued polishing his ideas until he was finally
released on April 29, 1832.
A month after his release, on
May 30, was
Galois' fatal duel. The true motives behind this duel that ended
his life will most likely remain forever obscure. There has been a
lot of speculation, much of it spurious, as to the reasons behind
it. What is known is that five days before his death he wrote a
letter to Chevalier which clearly alludes to a broken love
affair.
Some archival investigation on the original letters reveals that
the woman he was in love with was apparently a certain Mademoiselle
Stéphanie-Felicie
Poterin du Motel, the daughter of the physician at the hostel
where Galois remained during the final months of his life.
Fragments of letters from her copied by Galois himself (with many
portions either obliterated, such as her name, or deliberately
omitted) are available. The letters give some intimation that Mlle.
du Motel had confided some of her troubles
with Galois, and this might have prompted him to provoke the duel
himself on her behalf. This conjecture is also supported by some of
the other letters Galois later wrote to his friends the night
before he died. Much more detailed speculation based on these scant
historical details has been interpolated by many of Galois'
biographers (most notably by
Eric
Temple Bell in
Men of
Mathematics), such as the oft-repeated conjecture that the
entire incident was stage-managed by the police and royalist
factions to eliminate a political enemy.
As to his opponent in the duel, Alexandre Dumas names Pescheux
d'Herbinville, one of the nineteen artillery officers on whose
acquittal the banquet that occasioned Galois' first arrest was
celebrated and
Du Motel's fiancee. However,
Dumas is alone in this assertion, and extant newspaper clippings
from only a few days after the duel give a description of his
opponent which is inconsistent with d'Herbinville, and more
accurately describes one of Galois' Republican friends, most
probably Ernest Duchatelet, who was also imprisoned with Galois on
the same charges. Given the conflicting information available, the
true identity of his killer may well be equally lost to
history.
Whatever the reasons behind the duel, Galois was so convinced of
his impending death that he stayed up all night writing letters to
his Republican friends and composing what would become his
mathematical testament, the famous letter to Auguste Chevalier
outlining his ideas.
Hermann Weyl, one
of the greatest mathematicians of the 20th century, said of this
testament, "This letter, if judged by the novelty and profundity of
ideas it contains, is perhaps the most substantial piece of writing
in the whole literature of mankind." However, the legend of Galois
pouring his mathematical thoughts onto paper the night before he
died seems to have been exaggerated. In these final papers he
outlined the rough edges of some work he had been doing in analysis
and annotated a copy of the manuscript submitted to the academy and
other papers. On 30 May 1832, early in the morning, he was shot in
the
abdomen and died the following day
at ten in the Cochin hospital (probably of
peritonitis) after refusing the offices of a
priest. He was 20 years old. His
last
words to his brother Alfred were:
Ne pleure pas, Alfred ! J'ai besoin de
tout mon courage pour mourir à vingt ans ! (Don't cry, Alfred!
I need all my courage to die at twenty.)
Much of the drama surrounding the legend of his death has been
attributed to one source, Eric Temple Bell's
Men of
Mathematics.
Galois' mathematical contributions were published in full in 1843
when
Liouville reviewed his
manuscript and declared it sound. It was finally published in the
October–November 1846 issue of the
Journal
de Mathématiques Pures et Appliquées. The most famous
contribution of this manuscript was a novel proof that there is no
quintic formula, that is, that
fifth and higher degree equations are not solvable by radicals.
Although
Abel had already
proved the impossibility of a "quintic
formula" by radicals in 1824 and
Ruffini had published a solution in 1799 that
turned out to be flawed, Galois' methods led to deeper research in
what is now called
Galois Theory. For
example, one can use it to determine, for
any polynomial
equation, whether it has a solution by radicals.
Contributions to Mathematics
Unsurprisingly, Galois' collected works amount to only some 60
pages, but within them are many important ideas that have had
far-reaching consequences for nearly all branches of mathematics.
His work has been compared to that of
Niels Henrik Abel, yet another
mathematician who died at a very young age, and much of their work
had significant overlap.
Algebra
While many mathematicians before Galois gave consideration to what
are now known as
group, it was
Galois who was the first to use the word 'group' (in French
groupe) in the technical sense it is understood today,
making him among the key founders of the branch of algebra known as
group theory. He developed the concept
that is today known as a
normal
subgroup. He called the decomposition of a group into its left
and right
cosets a 'proper decomposition' if
the left and right cosets coincide, which is what today is known as
a normal subgroup. He also introduced the concept of a
finite field (also known as a
Galois field in his honor), in essentially the
same form as it is understood today.
Galois Theory
Galois' most significant contribution to mathematics by far is his
development of Galois theory. He realized that the algebraic
solution to a
polynomial equation is
related to the structure of a group of
permutations associated with the roots of the
polynomial, the
Galois group of the
polynomial. He found that an equation could be solvable in
radicals if one can find a series of normal
subgroups of its Galois group which are
abelian, or its Galois group is
solvable. This proved to be a fertile
approach, which later mathematicians adapted to many other fields
of mathematics besides the
theory of
equations which Galois originally applied it to.
Analysis
Galois also made some contributions to the theory of
Abelian integrals and
continued fractions.
See also
Notes
References
- Laura Toti Rigatelli, Evariste Galois, Birkhauser,
1996, ISBN 3764354100. This biography challenges the common myth
concerning Galois' duel and death.
- Ian Stewart,
Galois Theory, Chapman and
Hall, 1973, ISBN 0412108003. This comprehensive text on Galois
Theory includes a brief biography of Galois himself.
- The Equation That Couldn't Be Solved: How Mathematical
Genius Discovered the Language of Symmetry by Mario Livio, Souvenir Press 2006, ISBN
0-285-63743-6
- Leopold Infeld. Whom the Gods
Love: The Story of Evariste Galois. Reston,
Va.: National Council of
Teachers of Mathematics, 1948. ISBN 0873531256. Classic
fictionalized biography by physicist Infeld.
- Jean-Pierre Tignol. Galois's theory of algebraic
equations. Singapore: World Scientific, 2001. ISBN
981-02-4541-6. Historical development of Galois theory.
- Alexandre Astruc, Evariste Galois, Flammarion, 1994,
ISBN 2-08-066675-4 (in French)
External links