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At 16, Galois took the examinations to enter the prestigious Poly technique -- and failed. Years later Terquem remarked, "A candidate of superior intelligence is lost with an examiner of inferior intelligence." However, Galois found a mathematics teacher, Louis Richard, and really started studying and doing mathematics. His first paper, on continued fractions, was published when he was 17.
At 18, Galois reapplied to the Poly technique, and again the examination went badly. Finally, during the oral part of the exam, he lost patience with one of the examiners and threw the eraser at him. It was a hit, but Galois could never apply there again.
At 19, Galois attended the university and wrote three original papers on the theory of algebraic equations. He submitted them to the Academy of Sciences for the competition in mathematics. The Secretary of the Academy took them home to read, but then died before writing a report about them and the papers were never found. Galois was understandably upset: "Genius is condemned by a malicious social organization to an eternal denial of justice in favor of fawning mediocrity."
In 1830 the French masses revolted, and Galois was a staunch supporter. The director of the school locked the students in the school during the fighting and then expelled Galois for a public letter he wrote condemning the director. Galois tried to start his own school of mathematics, but got no students, so he joined the National Guard -- "If a carcass is needed to stir up the people, I will donate mine." 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. "Thus it happened that he experienced his one and only love affair. In this, as in everything else, 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.
Galois was shot in the intestines, and was taken to the hospital. He comforted his brother with "Don't cry, I need all my courage to die at twenty." He died the day after the duel and was buried in an unmarked, common grave.
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 is essentially the "complete" theory of the roots of polynomial equations in one variable. That means, it presents as complete a picture as possible in the general case of the solutions of polynomial equations. The study of such equations is one of the oldest parts of mathematics. The formulas for solutions of a quadratic equation, and sometimes a cubic equation, are taught in secondary schools. There are also explicit formulas for quartic equations, but not for quintics and equations of higher degree. Basic geometric problems like construction with ruler and compass of regular polygons and trisection of angles can be interpreted in terms of Galois theory (and thereby classified as solvable or not).
Galois theory is hardly a new part of mathematics. It is, like most things, the work of many people, but the most important ideas and results were conceived by Évariste Galois in 1832. In modern terminology, 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. 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.
In Galois theory, 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.
We can introduce symbols for roots of certain equations, e. g. the square route of two and i, and then express other roots in terms of those symbols. It turns out that when one adds such symbols to a field (i. e. "adjoins" them) and uses the equation they satisfy as an additional axiom, then the enlarged set also satisfies all the axioms for a field, and it is called an extension field.
Looking again at any polynomial equation, one finds that it can have at most in roots in any extension field, where n is the degree of the polynomial. It may have fewer distinct roots if some are repeated: x2 + 2x + 1 = (x+1)2 = 0 has just a single root (x = -1).
Galois' brilliant insight was that one can know essentially "everything" there is to know about the roots of polynomial equations by considering a new object, a group, namely the group of all "reasonable" permuations of those roots. Here, "reasonable" is not a technical term.
Galois concluded that not all permutations of the roots of a polynomial may be reasonable. This may happen if there are polynomial relationships among the roots with coefficients in the base field.
For instance, in the polynomial:
| f(x) = (x-i)(x+i)(x-2i)(x+2i) = x-4 + 5x2 + 4 |
The Galois group is a way of encoding all available information about the relationships of the roots of polynomials with coefficients in the base field that factor completely in the extension field. So in order to study all roots of a given polynomial, it is sufficient to find an extension field that contains all of the roots and examine the Galois group.
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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:
This project was prepared and presented by students: Mark Walker and Nate Hiers in 1999.