Rebels

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 Legendres 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 16th century the Italians del Ferro,
  Tartaglia and Cardano solved x3 and later
  Ferrari x4
• 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 (1765- 1822)

• And the winner is..
• Niels Henrik Abel (1802- 1829)

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 Abels 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

• 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

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.

t3
c1
c2
t2
I
t1
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
o I c1 c2 t1 t2 t3
I I c1 c2 t1 t2 t3
c1 c1 c2 I t3 t1 t2
c2 c2 I c1 t2 t3 t1
t1 t1 t2 t3 I c1 c2
t2 t2 t3 t1 c1 I c1
t3 t3 t1 t2 c1 c2 I
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 Loyds Challenge 100 000 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 Johns 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 rotations 120o, 240o, 360o
3 mirror reflections
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 Of eq Polyhedra of symmetry elements in polyhera
2 ----- 2! 2
3 equilateral 3! 6
4 tetrahedron 4! 24
5 icosahedron 5! 120
15
Galois Magic

• 1. Showed that every equation has its 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 its Galois group S5 has one of
  its subgroups of size 60 which is not a prime.
  Therefore its Galois group is of the wrong type
  and the equation cannot be solvable by formula.

No of objects 5 6 7 8 9 10
No. of even permutations 60 360 2500 20 160 181 440 1 814 400
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 Fermats
  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 Loyds 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

1. Expand 2. and equate coefficients to get

Equation coefficients from 1. and 2.
Given the general quadratic solution