Transcript Evarist Galois and Group Theory

Rebels
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Evariste Galois and the
Quintic
Pierre Cuschieri
Math 5400
Feb 12, 2007
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Evariste Galois :Biography
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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.
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Solvability of equations
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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
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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.
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Getting closer and closer…
Paolo Ruffini (1765- 1822)
Equations with n ≥ 5 not solvable
by a simple formula. Treatise
was difficult to follow and not
regarded highly.
… And the winner is…..
Niels Henrik Abel (1802- 1829)
The first to successfully
prove that the equation of
fifth degree was not
solvable algebraically.
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Galois and the quintic
 Recall Abel’s
 BIG
work on quintic
? : 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.
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Galois quote
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“ 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”
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..and so Galois continued where Lagrange left
off in the solvability of algebraic equations
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The language of Symmetry
Group Theory- Basics
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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
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Permutation of a group
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Permutation is an arrangement of elements in group( even or odd ).
Ex: consider all possible permutations of the letters
a,b,c.
a b c a b c


 
 a b c a c b
I
t1
a b c a b c

 

c b a b a c
t2
Operations: I = identity,
t3
a b c


 c a b
c1
t = transposition,
a b c


b c a
c2
c = cyclic
Each operation can be regarded as a member of a group
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Multiplication Table for the six permutations
o
I
c1
c2
t1
t2
t3
Where operations:
I
I
c1
c2
t1
t2
t3
c1
c1
c2
I
t2
t3
t1
t = transpose,
c2
c2
I
c1
t3
t1
t2
t1
t1
t3
t2
I
c1
c1
t2
t2
t1
t3
c1
I
c2
c = cyclic,
t3
t3
t2
t1
c2
c1
I
O
= “followed by”
Ex c1o t1 = t2 means transformation 1 “followed by “cyclic 1 yields transformation 2
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Even vs Odd permutations
Sam Loyd’s Challenge
$100 000 to anyone who
can interchange the numbers
14 and 15 while keeping
Everything else the same
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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
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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
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Summary of isomorphic properties of
algebraic equations and polyhedra
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I. Properties of the symmetry groups of algebraic
equations correspond or can be visualized by
comparing them to polyhedra
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Degree
Of eq’
Polyhedra
2
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2! = 2
3
equilateral
3! = 6
4
tetrahedron
4! = 24
5
icosahedron
5! = 120
# of symmetry
elements in
polyhera
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Galois Magic
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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 )
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2. Defined the concept of a normal subgroup
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3. Tried to deconstruct these groups into simpler ones
called prime cyclic groups. If this was possible, then the
equation was solvable by formula.
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Fate of the quintic
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For the quintic it’s Galois group S5 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
5
6
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No. of even
permutations
60
360 2500
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9
20 160 181 440
10
1 814 400
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At turn for the worse
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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.
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Group Theory after Galois
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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
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GT and High School Math
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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.
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Last words by….
Greek Poet Menander ( 300 BC )
” Those who are beloved by the
Gods die young ”
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Appendix A
x2 
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b
c
x
0
a
a
( x  x1 )( x  x)  0
Symmetry of the Quadratic
1. Divide general quadratic by a
2.
Solution using putative roots
x 2  ( x1  x 2 ) x  x1 x2  0
x1  x2 
b
c
...and ..x1 x2 
a
a
3. Expand 2. and equate coefficients to get:
Equation coefficients from 1. and 2.
given
Given the general quadratic solution
 b  b 2  4ac
x1, 2 
2a
sub...in..values... for...a, b, c....to..get
1
( x1  x2 )  ( x1  x2 )  4 x1 x2
2
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which..is...symmetric...with..x1..x2 ..transposed
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