October 25, 1811 to October 25, 2011
Primary References
The Question
The Equation:
cn x  cn1x
n
n1
 cn2 x
n 2
  c3 x  c2 x  c1x  c0  0
3
2
An algebraic solution uses:
Ancoefficients
algebraic
is also
• The
cn, cnsolution
– 1, …, c2, c1, c0,
sometimes
called
“solving
by
• Addition and subtraction, + - ,
radicals.”
•Multiplication and division, x / , and
• Extraction of roots,
n
.
A Familiar Example
The Question:
ax  bx  c  0 ?
2
The Answer:
 b  b  4ac
x
.
2a
2
An Unfamiliar Example
The Question:
ax  bx  cx  d  0 ?
3
2
The Answer:
2b
b 1 1
x1    3  2b3  9abc  27a 2 d 
3a 3a 2 

x2  

1 1 3
2
3
 2b  9abc  27a d 
3a 2 
2b

2b
2b
3
b 1  3 1  3
2
3

 2b  9abc 27a d 
3a
6a
2
1  3 1  3
2
3

2
b

9
abc

27
a
d

6a
2



 9abc  27a d  4 b  3ac 

2
2
3
2


2
3
 9abc  27a 2 d  4 b 2  3ac 

b 1  3 1  3
2
3

 2b  9abc 27a d 
3a
6a
2
1  3 1  3
2
3
 2b  9abc  27a d 
6a
2
x3  
3
3
2b



2
3
 9abc 27a 2 d  4 b2  3ac 




2
3
 9abc  27a 2 d  4 b 2  3ac 

2b
3
3
3



2
3
 9abc 27a 2 d  4 b 2  3ac 




 9abc  27a d  4 b  3ac 

2
2
2
3
The Question:
ax  bx  cx  dx  e  0 ?
4
3
2
The Answer: There is an algebraic equation!
The Question:
ax  bx  cx  dx  ex  f  0 ?
5
4
3
2
The Answer: There is an no algebraic equation!
A Brief History of
Equations:
The Quadratic
Tigris
Euphrates
Babylon
NASA image from:
http://www.grabovrat.com/mapsViews/mapsViewsFig/mapsViews815.jpg
NASA image from:
http://www.grabovrat.com/mapsViews/mapsViewsFig/mapsViews815.jpg
Nile River
Egypt
From Wikipedia
Brahmagupta (598-670)
The product or ratio
of two debts is a
fortune; the product
or ratio of a debt and
a fortune is a debt.
http://www.fathersofmathematics.com/
Muhammad ibn Musa
al-Khwarizmi (circa 790-850)
Kitab al-jabr we
al-muqabalah
(The Condensed
Book on Restoration
and Balancing)
Abraham bar Hiyya Ha-nasi (1070-1136)
 b  b 2  4ac
x
2a
http://www.esefarad.com/?p=8536
A Brief History of
Equations:
The Cubic
• 1515, Scipione dal Ferro's (1465-1526)
ax  bx  c
3
(Doesn’t publish; tells his student Antonio
Maria Fiore.)
• 1535, Fiore challenged Niccolo Tartaglia to
a “cubic solving contest.”
Niccolo Tartaglia (1500-1557)
A Brief History of
Equations:
The Quartic
• 1540, Lodovico Ferrari (1522 – 1565)
solves the quartic equation:
ax  bx  cx  dx  e  0.
4
3
2
Gerolamo Cardano (1501-76)
http://nsm1.nsm.iup.edu/gsstoudt/history/image
s/arsmagna.html
A Brief History of
Equations:
The Quintic
Unsuccessful Attempts
•Scot James Gregory (1638-75)
•Ehrenfried Walther von Tschirnhaus
(1651-1708)
•Etienne Bezout (1730-83)
•Erland Samuel Bring (1736-98)
Paolo Ruffini (1765-1822)
Niels Henrik Abel 1802 - 1829
The Revised Question
When is there an algebraic equation
which gives the solutions of a
polynomial equation?
Evariste Galois
1811-1832
(1866)
Bourg-la-Reine
http://www.zonu.com/fullsize-en/2009-09-18-10505/Environs-of-Paris-Map-France-1866.html
France during Galois’ Life
http://eurohist.tumblr.com/post/281851262
1/napoleon-bonaparte-how-people-usuallythink-of
http://mateengreenway.com/steamp
unk/Space1889timeline.htm
Charles X
From Wikipedia
Left his parents house
for the Lycee Loiusle-Grand, 1823.
October 25, 1811
1816
1821
1826
May 30, 1832
Lycee Louis-le-Grand
http://www.galois.ihp.fr/
From Wikipedia
Adrien-Marie Legendre (1752 – 1833)
http://www.ee.nthu.edu.tw/~sdyang/C
ourses/PDE.htm
http://mathdl.maa.org/mathDL/46/?
pa=content&sa=viewDocument&n
odeId=2591&bodyId=3665
Louis-Paul-Emile Richard (1795-1849)
http://www.wpclipart.com/people/faces/anonymous/photo_not_available_large.png.html
Galois publishes his
first paper: Proof of a
Theorem on Periodic
Continued Fractions,
April 1829.
October 25, 1811
1816
1821
1826
May 30, 1832
The Two Questions Again
1. Find all the equations of a specific
degree that are solvable by algebra
(i.e., by radicals), and
2.Decide whether a given equation can
be solved by algebra or not.

Two memoires.
The French Academy of Sciences 1
Jean Baptiste
Joseph Fourier
(1768 – 1830)
Augustin Louis
Cauchy
(1789 – 1857)
Claude Louis
Marie Henri
Navier
(1785 – 1836)
The French Academy of Sciences 2
Augustin Louis
Cauchy
(1789 – 1857)
Siméon Denis Poisson
1781 - 1840
Evariste’s father,
Nicolas-Gabriel,
commits suicide,
July 2, 1829.
October 25, 1811
1816
1821
1826
May 30, 1832
The Ecole Polytechnique
http://www.galois.ihp.fr/
Galois fails the
entrance exam to the
Ecole Polytechnique,
June 1828 & July 1829.
October 25, 1811
1816
1821
1826
May 30, 1832
Galois enters Ecole
Preparatoire, February
1830; Meets Auguste
Chevalier there.
October 25, 1811
1816
1821
1826
May 30, 1832
?
Augustin Louis
Cauchy
(1789 – 1857)
Grand Prix de Mathematiques
X
Jean Baptiste
Joseph Fourier
(1768 – 1830)
X
A Third Try with the Memoire
Siméon Denis Poisson
1781 - 1840
“We made all possible
efforts to understand M.
Galois' evidence. His
thesis is neither clear
enough, nor sufficiently
developed to enable us to
judge its rigour. Neither
are we able to provide a
clear idea of this work. For
this reason, we return your
manuscript in the hope that
you will find M. Poisson's
observations of use for
your future research.”
Galois’ Five Publications
1. “Proof of a Theorem on Periodic Continued Fractions” in Annals of Pure
and Applied Mathematics, 19, 294-301 (April 1829).
2. “An Analysis of the Resolution of Algebraic Equations” in Bulletin of the
Mathematical, Physical, and Chemical Sciences, 13, 271-272 (April 1830).
3. “Note on the Resolution of Numerical Equations” in Bulletin of the
Mathematical, Physical, and Chemical Sciences, 13, 413-414 (June
1830).
4. “On the Theory of Numbers” in Bulletin of the Mathematical, Physical,
and Chemical Sciences, 13, 428-435 (June 1830).
5. “Notes on Some Points of Analysis” in Annals of Pure and Applied
Mathematics, 11, 182-184 (December 1830).
WARNING: Translations are due to Dr. Bob!
More
Politics
Charles X
• Dissolved Parliament
• Suspended freedom of
press
• Annulled recent
elections
• Introduced a property
qualification for voting
From Wikipedia
Letter to a Newspaper
http://abcteach.com/directory/clip_art/miscellaneous/
Society of Friends
http://www.wpclipart.com/people/faces/anonymous/photo_not_available_large.png.html
Louis-Phillipe I
• Dismissed the
popular General
Lafayette
• Disbanded the
National Guard.
From Wikipedia
The Trial of the Nineteen
http://etc.usf.edu/clipart/21900/21925/trial_21925.htm
The Banquet
http://www.menshealth.co.uk/lose-weight/burn-fat/three-ways-to-lose-weight-fast
Galois in Court
This is what happened: I had a knife which I had
used to cut meat during the meal. I brandished it
saying ‘To Louis-Phillipe, if he betrays us!’
These words were only heard by the people next
to me, considering the whistling that had begun
after the beginning of my utterance, because
people thought I was proposing a toast to the
good health of Louis-Phillipe.
Re-Arrest
Galois is in jail from
July 1831 to April 1832.
October 25, 1811
1816
1821
1826
May 30, 1832
Sainte-Pelagie Prison
http://www.parisrevolutionnaire.com/spip.php?article503
Stephanie Poterin-Dumotal
http://fineartamerica.com/featured/french-victorian-girl-jonny-petros.html
Letters (May 29, 1832)
(1) To all republicans: “I beg my patriotic friends not to chide me for
(2)
dying in any other way than for my country. I die, the victim of
a cruel coquette, and of two of her victims. My life fades away
amidst trivial gossip. … Adieu! Life was dear to me, for the
common good. Pardon for those who killed me. They were
acting in good faith.”
“I have been challenged to a duel by two patriots… I cannot
refuse.”
(3) To Auguste Chevalier:
A brief summary of the memoire
deposited at the Academy, adding some new theorems and
conjectures covering seven pages; he concluded: “I do not have
enough time and my ideas are not sufficiently well developed in
this area, which is enormous.”
The Duel
http://weetlogs.scilogs.be/gallery/11/previews-med/Galois.jpg
May 31, 1832
http://www.indianhospitalsguide.com/cochin-hospital-paris-france/
Myths
Laura Rigatelli
The duel was
arranged to provide a
body around which
the republicans could
circle and then start
an uprising against
the monarchy.
Mario Livio
The duel was arranged
by the uncles of
Stephanie PoterinDumotal. She was the
“infamous coquette”
mentioned in Galois’
first letter.
Galois Theory
Three Geometry Problems
1.Squaring the circle: Drawing a square
the same area as a given circle.
2.Doubling the cube: Drawing a cube with
twice the volume of a given cube.
3.Trisecting the angle: Dividing a given
angle into three smaller angles all of the
same size.
Modern Galois Theory
The polynomial equation p(x)=0 is solvable by
radicals if and only if the Galois group of p is
solvable.
Groups, normal
subgroups, fields,
extension fields, …
http://www.galois.ihp.fr/en/
Evariste Galois
Born: October 25, 2011
References
Quadratic, Cubic, and Quartic Equations, (http://wwwgroups.dcs.st-and.ac.uk/~history/HistTopics/
Quadratic_etc_equations.html)
Uncredited images of mathematicians are from http://wwwgroups.dcs.st-and.ac.uk/~history/
Livio, M. The Equation That Couldn’t Be Solved, NY: Simon &
Schuster (2005).
Rigatelli, L.T., Evariste Galois 1811-1832, translated by J.
Denton. Boston: Birkhauser Verlag (1996)
Je n’ai pas le temps!
Bicentennial of his birth:
October 25, 1811 – October 25, 2011
Theorem. The polynomial equation
p(x)=0 is solvable by
radicals iff the Galois
group of p is solvable.
A letter explaining his theory.
May 29, 1832.
http://www.sciencephoto.co
m/media/363863/view
http://langevin.univ-tln.fr/NOTES/GALOIS/Galois.7.jpg
October 25, 1811
1816
1821
1826
May 30, 1832
Saint Evariste
http://magnificat.ca/cal/fran/10-26.htm
http://140.111.115.8/longlife/mathstamp/galois.jpg
http://www.white-rabbit.jp/Column/essay6.html
http://www.white-rabbit.jp/Column/essay6.html
http://www.white-rabbit.jp/Column/essay6.html
http://www.white-rabbit.jp/Column/essay6.html
http://www.white-rabbit.jp/Column/essay6.html
http://www.white-rabbit.jp/Column/essay6.html
ax  bx  cx  d  0
3
 b
bc
d 
x  
 2   
3
2a 
 27 a 6a
3
3
 b

bc
d
 3 
 2   
3
2a 
 27 a 6a
3
2
2
 b
bc
d   c
b 

 2      2 
3
2a   3a 9a 
 27 a 6a
3
2
2
3
3
 b
bc
d   c
b 
b




 2     2  
3
2a   3a 9a 
3a
 27 a 6a
3
2
ax  bx  cx  d  0
3
x1  
2
b 1 1 3
 3  2b  9abc  27a 2 d 
3a 3a 2 
1 1 3
 3  2b  9abc  27a 2 d 
3a 2 
2b
3
2b
3



2
3
 9abc  27a 2 d  4 b 2  3ac 




 9abc  27a d  4 b  3ac 

2
2
2
3
b 1 i 3 1  3
2
3
x2   
 2b  9abc  27a d 
3a
6a
2
1 i 3 1  3
2
3

2
b

9
abc

27
a
d

6a
2
x3  

2b
2b
3
b 1 i 3 1  3
2
3

 2b  9abc 27a d 
3a
6a
2
1 i 3 1  3
2
3
 2b  9abc  27a d 
6a
2
2b



 9abc  27a d  4 b  3ac 

2

2

3
2

 9abc 27a d  4 b  3ac 

2
2b
3
3
3
2

3
2


2
3
 9abc 27a 2 d  4 b 2  3ac 




2
3
 9abc  27a 2 d  4 b 2  3ac 

2b
b 1 1
x1    3  2b3  9abc  27a 2 d 
3a 3a 2 

x2  

1 1 3
2
3
 2b  9abc  27a d 
3a 2 
2b

2b
2b
3
b 1 i 3 1  3
2
3

 2b  9abc 27a d 
3a
6a
2
1 i 3 1  3
2
3

2
b

9
abc

27
a
d

6a
2



 9abc  27a d  4 b  3ac 

2
2
3
2


2
3
 9abc  27a 2 d  4 b 2  3ac 

b 1 i 3 1  3
2
3

 2b  9abc  27a d 
3a
6a
2
1 i 3 1  3
2
3
 2b  9abc 27a d 
6a
2
x3  
3
3
2b



2
3
 9abc  27a 2 d  4 b 2  3ac 




2
3
 9abc 27a 2 d  4 b 2  3ac 

2b
3
3
3



2
3
 9abc 27a 2 d  4 b 2  3ac 




 9abc  27a d  4 b  3ac 

2
2
2
3
Saint Andrews, Tartaglia
http://www.swaen.com/antique-map-of.php?id=15378