Transcript Slide 1

• Evariste Galois
(father of modern
algebra)
• Born: 25 Oct., 1811,
in Bourg La Reine
(near Paris),
France
• Died: 31 May, 1832,
in Paris, France
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Childhood
• Bourg-la-Reine is about 10 km south of
Paris.
• Family renowned for school which dates
back to revolution
• Parents are well-educated in all subjects
considered important at that time : classical
literature, religion and philosophy
• No record of explicit family talent on
mathematics, but nothing on the contrary
either!
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Childhood
• He inherited composing rhymed couplets
to amuse friends and family
• Attended a college in Reims at the age of
10.
– Mother change of mind and home-schooling
• Educated in Latin, Greek, Rhetoric
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School
• 1823, age 12. Louis-le-Grand Lycee in
Paris
• Turbulent times of unbalanced political
views – church, royalist, and republicans
• Rebellion at the first year of school
• Uncertainity about his school years.
Admitted to fourth class (sixth class being
the first year and the first class being the
last year at school).
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School
• 1825-26 downturn on performance due to
earache and repeat second class
• Feb 1827, first mathematics class with M.
Vernier.
• Legendre’s text on geometry, and get
acquainted w/ theory of equations by
Lagrange’s works.
• Poor on lessons other than mathematics
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School
• "Whats's dominating him is the fury of mathematics;
also, I think that it would be better for him if his parents
would agree to let him study solely mathematics. He is
wasting his time here and he does nothing but torment
his teachers and by doing so heaps punishments on
himself."
• "His fast apprehension is a legend by now, in which soon
nobody will believe; there is a trace of particularness and
of carelessness in his homework, if he does it at all; he is
constantly busy with things, which he does not have to
do, but is affecting him."
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School
• Attempted to enter Ecole twice for attaining best possibilities in
mathematics
– 1829, failed the exam
• Falsified scandal of his fathers poems led them out of Bourg-laReine to Paris
– This scandal has ended with his father’s suicide on July, 2nd.
• Took the entrance exam 2nd time. (legendary).
– Fatal question: Describe the theory of arithmetic logarithms
– Fatal answer: Why not ask theory of logarithms, there is no such
thing like arithmetic logarithms?
– The Result : Fail again!
• Bell reports that : One of the examiners had discussed both falsely and stubbornly a
mathematical fact. In a fury and despair he hurled the sponge into the face of his tormentor.
Twenty years later we find In the Nouvelles Annales Mathématiquesthis: "a candidate of superior
intelligence was ruined by an examiner of minor intelligence.
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politics
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At 19, Galois attended the university and wrote three original papers on the theory of algebraic
equations.
In 1830 supported the French revolution.
Director expelled Galois for a public letter he wrote condemning the director.
Galois was jailed for supposedly threatening the King, but was found 'not guilty' by a jury. Finally
he was convicted and sentenced to 6 months in jail for "illegally wearing a uniform."
When he was finally released, his last misadventure began. “His one and only love affair.
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he was unfortunate. Galois took it violently and was disgusted with love, with himself, and with his girl."
A few days later Galois encountered some of his political enemies and "an affair of honor," a duel,
was arranged. Galois knew he had little chance in the duel, so he spent all night writing the
mathematics which he didn't want to die with him, often writing "I have not time. I have not time."
in the margins.
He sent these results as well as the ones the Academy had lost to his friend Auguste Chevalier,
and, on May 30, 1832, went out to duel with pistols at 25 paces.
Twenty four years after Galois' death, Joseph Liouville edited some of Galois' manuscripts and
published them with a glowing commentary. "I experienced an intense pleasure at the moment
when, having filled in some slight gaps, I saw the complete correctness of the method by which
Galois proves, in particular, this beautiful theorem: In order that an irreducible equation of prime
degree be solvable by radicals it is necessary and sufficient that all its roots be rational functions
of any two of them."
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Galois Theory
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Introduced the idea of Fields with Norwegian mathematician Niels Henrik Abel during
their studies of the roots of polynomials.
Galois theory is the complete theory of roots of polynomial equations in one variable.
It is formulated using the concept of an algebraic structure called a field.
A field is a set, which may be finite or infinite, that has two distinct but closely related
group structures on it.
The most common examples are the rational numbers, Q, the real numbers, R, and
the complex numbers C. Each of these has group structures corresponding to the
operations of addition and multiplication.
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The two operations are related in that multiplication is required to be "distributive" with
respect to addition, i. e. a(b + c) = ab + ac. Everything else follows from the group axioms
and the distributive rule.
The primary object of interest is the polynomial equation in one variable, where the
coefficients {a-k} are all in some specific "base" field.
The goal of the theory is to say as much as possible about the roots of such
equations, that is, values of x for which the equation is true. In general, the roots of
the equation will not be members of the same base field as the coefficients. One may
think of the roots simply as abstract objects which can be "adjoined" to the base field
to provide solutions of the equation.
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Conclusion
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The 5 Questions
1) What did Everiste Galois discover / prove?
– A) The 'complete' theory of the roots of polynomial equations in one variable
2) In what ways do we still use his theorem today?
A) His theory formed some of the basics of geometry such as square root, constructions, and polynomial
equations.
3) Why did he only have one famous equation in mathematics?
A) Gaining many enemies in politics and a duel was arranged. Galois knew he had a very slim chance in the
duel, so he spent all night writing the mathematics which he didn't want to die with him, often writing "I have not
time. I have not time." in the margins.
4) What was his most famous equation about?
A) Galois invented group theory while trying to solve this problem - the problem was, can you find a formula
like the famous quadratic formula that finds the roots of a fifth degree polynomial? Formulas were known at this
time for all polynomials of degree 3 or four, but there was no general method for finding roots of higher order
polynomials. Galois proved that no such general method could be found, at least using a purely algebraic
formula. The traditional accounts claim that he figured this all out in his head and only wrote it down in haste one
night before a duel.
5) What other things did his theory help with?
A) There are a few other things that his theories had helped with.
These are:
the impossibility of trisecting the angle
duplicating the cube
squaring the circle
solving equations of the fifth degree
the possibility of constructing regular polygons with 17, 257, or 65537 sides
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