1 Rebels

2 Evariste Galois and the Quintic Pierre Cuschieri Math 5400 Feb 12, 2007

3 Evariste Galois :Biography  Born 1811 in Bourge-la-Reine France.  Father was Nicolas Gabriel Galois.  Attended school: Louis-le-Grand in Paris.  Regarded as odd and quiet.  Read Legendre’s “Elements of Geometry”  Failed entrance to ecole Polytechnique in June 1826  Attends ecole Normal and meets Mr Richard who recognizes Galois as a genius.  He is encouraged to send his work to Cauchy who misplaces it.  In 1829 he published his first paper on continued fractions.  Became interested in solving the quintic based on the works of Lagrange.

4 Solvability of equations  6000 yrs ago linear equations where solvable  Babylonians ( 4000 yrs ) some types of quadratics  Greeks  quadratics using ruler and compass  By the 16 th century the Italians del Ferro, Tartaglia and Cardano solved … x 3 and later Ferrari … x 4   IT STOPS HERE !  Lagrange, Euler, and Leibniz unsuccessful at solving the quintic.  Algebra fails here !. It would take about 300 years later for the quintic dilemma to be answered.

5 Getting closer and closer… Paolo Ruffini ( ) … And the winner is….. … And the winner is….. Niels Henrik Abel ( ) Equations with n ≥ 5 not solvable by a simple formula. Treatise was difficult to follow and not regarded highly. The first to successfully prove that the equation of fifth degree was not solvable algebraically.

6 Galois and the quintic  Recall Abel’s work on quintic  BIG ? : How does one determine whether any given equation is solvable by a formula or not?  Galois: studied equations from a different perspective; he looked at the permutation symmetry of the roots, to determine the solvability of equations  group theory.

7 Galois quote Galois quote  “ jump on calculations with both feet; group the operations, classify them according to their difficulty and not according to their form; such according to me is the task of future geometers; such is the path I have embarked on in this work” ..and so Galois continued where Lagrange left off in the solvability of algebraic equations

8 The language of Symmetry Group Theory- Basics  Group- collection or set of elements together with an inner binary operation ( “multiplication”), satisfying the following rules or properties:  1. Closure  2. Associative  3. Identity  4. Inverse  Ex of groups: Set of I, {…-3,-2,-1,0,1,2,3…} Operations involved can be as simple as +,-,x, / to complicated symmetry transformations such as rotations of a fixed body Operations involved can be as simple as +,-,x, / to complicated symmetry transformations such as rotations of a fixed body

9 Permutation of a group  Permutation is an arrangement of elements in group- ( even or odd ).  Ex: consider all possible permutations of the letters a,b,c. t2 t3 c1c2 It1 Operations: I = identity, t = transposition, c = cyclic Each operation can be regarded as a member of a group

10 Multiplication Table for the six permutations oIc1c2t1t2t3 IIc1c2t1t2t3 c1c1c2It3t1t2 c2c2Ic1t2t3t1 t1t1t2t3Ic1c2 t2t2t3t1c1Ic1 t3t3t1t2c1c2I Where operations: t = transpose, c = cyclic, O = “followed by” Ex c1o t1 = t2 means transformation 1 “followed by “cyclic 1 yields transformation 2

11 Even vs Odd permutations Sam Loyd’s Challenge $ to anyone who can interchange the numbers 14 and 15 while keeping Everything else the same

12 SYMMETRY Galois studied the symmetry rather than the solutions which he treated like objects that could be interchanged with one another Examples of symmetry: 1. ab +bc +ca is symmetric under the cyclic permutation of a,b,c 2. Jack is John’s brother

13 Linking symmetry with permutations Permutation groups of roots of algebraic equations can be visualized by sets of symmetry operations on polyhedra.  symmetry point groups  Example : The group of 6 symmetries of an equilateral triangle is isomorphic to the group of permutations of three object a,b,c 3 mirror reflections 3 rotations 120 o, 240 o, 360 o

14 Summary of isomorphic properties of algebraic equations and polyhedra  I. Properties of the symmetry groups of algebraic equations correspond or can be visualized by comparing them to polyhedra  Degree Degree Of eq’ Polyhedra # of symmetry # of symmetry elements in polyhera elements in polyhera ! = 2 2! = 2 3equilateral 3! = 6 3! = 6 4tetrahedron 4! = 24 4! = 24 5icosahedron 5! = 120

15 Galois Magic  1. Showed that every equation has it’s own “symmetry profile”; a group of permutations now called Galois group, which are a measure of the symmetry properties of the equation. ( see appendix for example using the quadratic )  2. Defined the concept of a normal subgroup   3. Tried to deconstruct these groups into simpler ones called prime cyclic groups. If this was possible, then the equation was solvable by formula. 

16 Fate of the quintic  For the quintic it’s Galois group S 5 has one of it’s subgroups of size 60 which is not a prime. Therefore it’s Galois group is of the wrong type and the equation cannot be solvable by formula. No of objects No. of even permutations

17 At turn for the worse  Fails second attempt to ecole polytechnique but manages to publish papers on equations and number theory.  Galois begins to loose faith in the education and political situation and rebels.  Joins a revolutionary militant wing and ends up arrested  Falls in love while in a prison hospital but the affair is short lived  Challenged to a duel the day after his release.  Spends the entire night writing down his mathematical discoveries and gives them to Auguste Chevalier to hand over to Gauss and Jacobi.  The following day is shot in the duel and left for dead. CONSPIRACY?  Dies in the hospital the next day on May 31, 1832 from complications.

18 Group Theory after Galois  Charles Hermite in 1858 solved the quintic using elliptic functions  Arthur Cayley ( 1878 )- proved that every symmetric group is isomorphic to a group of permutations (ie, have the same multiplication table)  Felix Klein in 1884 showed relationship between the icosahedron and the quintic  GT is now used by chemists and physicists to study lattice structures in search of particles found in theory.  Used by Andrew Wiles to help him solve Fermat’s Last Theorem

19 GT and High School Math  Grade 9 - Measurement, classify objects in terms of their symmetry, define the types of symmetry, life story of a Math rebel.  Grade 10 - Math – introduction to quadratics  - demonstrate symmetry of quadratic and limitations of algebra.  Grade 11- – Symmetry in Functions and transformations and imaginary roots  Grade 12 - permutations, geometry, advanced functions, Sam Loyd’s puzzle.  Physics and Chemistry – demonstrate method of GT to finding particles.

20 Last words by…. Greek Poet Menander ( 300 BC ) ” Those who are beloved by the Gods die young ”

21 Appendix A Symmetry of the Quadratic  1. Divide general quadratic by a 2. Solution using putative roots 3.Expand 2. and equate coefficients to get: Given the general quadratic solution Equation coefficients from 1. and 2.