Evariste Galois ( father of modern algebra)

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 by Koray İNÇKİ

2. 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! by Koray İNÇKİ

3. 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 by Koray İNÇKİ

4. 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). by Koray İNÇKİ

5. 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 by Koray İNÇKİ

6. 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." by Koray İNÇKİ

7. 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-la-Reine 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. by Koray İNÇKİ

8. politics • 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. • 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." by Koray İNÇKİ

9. Galois Theory • 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. • 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. by Koray İNÇKİ

10. Conclusion • 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 by Koray İNÇKİ