Post on 24-Dec-2015
Fundamental Theorem of
Algebra A Cartoon-Assisted Proof
• Frank Wang• LaGuardia
Community College
Topics
• Motivation
• What Happens to Complex Numbers
• Complex Functions
• Fundamental Theorem of Algebra
• Abel’s Theorem
New York Times
Elusive Proof, Elusive Prover: A New Mathematical Mystery (Aug 15, 2006)
Letter to the New York Times
• In my modest attempt to understand Poincaré’s Conjecture discussed in the Times, I thought of how one of the greatest play ever written, Sophocles’s Oedipus Rex, enacts the notion of the sphere that can be expanded, reshaped, or contracted to a point. In teaching this play, I have used the image of the net to describe human fate: you can push at the boundaries of the net so that your individual choices allow you a personal identity, but you can never get outside the net—or the sphere… van Slyck
Popular Books
The most beautiful equation
1
sincos
i
i
e
ie
Julia set c=0.33+0.45 i
Mandelbrot set and bifurcation
• Mandelbrot set
}|)0(||{ ncQCcM
czzQc 2:
Newton’s Method in the Complex Plane
Differential Equations
dteeEeRyyL
dtettERyyL
LRttiti
LRt
/
/)cos(cos
Isaac Newton
• Invented Calculus• Laws of Motion• F=m a• Differential Equations
Galileo Galilei
• Pendulum Motion
l
g
Johannes Kepler
• Planetary Motion
02
)(2
2
rr
r
GMrr
(False) Conclusion
• Most systems of interest are periodic.
• Most differential equations are analytically solvable.
• Receive an A in Differential Equations, then one can solve any problem!
Henri Poincaré
• Three body problem can NOT be analytically solved
• The trajectory could be chaotic!
Sonya Kovalevskaya
Rigid body problems are NOT integrable except for
• Euler Case• Lagrange Case• Kovalevskaya Case
Kovalevskaya to Mittag-LefflerDear Sir,… It is a question of integrating
…, and I can show that these 3 cases are the only ones [integrable]…
• Roger Cooke Trans.
00
00
00
)(
)(
)(
xypqBArC
zxrpACqB
yzqrCBpA
Fundamental Quantum Condition
i
hqppq
2
In order to understand the Origin of the universe, we need to combine the General Theory of Relativity, with quantum theory. The best way of doing so, seems to be to use Feinman's idea of a sum over histories.
Richard Feynman
“The universe has every possible history.”
Richard Feinman was a colorful character, who played the bongo drums in a strip joint in Pasadena, and was a brilliant physicist at the California Institute of Technology. He proposed that a system got from a state A, to a state B, by every possible path or history.
A ~ eiS[g]/ћ
Sum over all metrics consistent with given boundary conditions
Feynman Sum Over Histories
Each path or history, has a certain amplitude or intensity, and the probability of the system going from A- to B, is given by adding up the amplitudes for each path. There will be a history in which the moon is made of blue cheese, but the amplitude is low, which is bad news for mice.
Fundamental Theorem of Algebra
Let p(z) be a polynomial over C, then there exists at least one z such that p(z)=0.
(Euler, Lagrange)
Gauss (1799)
Geometrical interpretation of complex numbers
iz
iz
iyxz
32
32
1
1
Complex Functions
xyyxv
yxyxu
yxivyxuzf
ixyyxiyxzf
zzf
2),(
),(
),(),()(
2)()(
)(
22
222
2
Complex Functions
13),(
3),(
),(),()(
)13(3)()(
)(
32
23
32233
3
yyxyxv
xyxyxu
yxivyxuzf
yyxixyxiiyxzf
izzf
z plane and w plane2)( zzfwz
z plane and w plane
izzfwz 3)(
z plane and w plane (Riemann’s Hypothesis)
)()( zzfwz
Complex NumbersCartesian vs Polar Forms
)tan,( 122
x
yyxr
rez
iyxzi
Parametrized curve
)2sin()2cos()( tittz
Map from z to w
)2sin()2cos()( tittz
izzw 3
De Moivre’s formula
)sin(cos
)sin(cos
ninrz
iriyxznn
Winding number (2)
zzw 01.02
)2sin()2cos()( tittz
Winding number (3)
zzw 05.03
)2sin()2cos()( tittz
Property
• When a loop is large enough, the winding number is the degree of the polynomial.
nw )(
Expanding r
izzzw 23
)2sin()2cos()( tirtrtz
Expanding r
izzzw 24
)2sin()2cos()( tirtrtz
Index 1, 2, 3
)(zfdt
dz
Index
10
)( 01
1
t
azatzdt
dz nn
n
Abel’s Theorem
If n>4 and f(x) is the general polynomial of degree n, then f(x) is not solvable by radicals.
Galois Theory
• The Galois group of the general polynomial is Sn, the symmetric group on n symbols.
• For n>4, Sn is not solvable.
Riemann surface
)(
)Re(
)Im(
)Re(
)(
5 zw
wh
zy
zx
zfw
Riemann surface
)(
)Re(
)Im(
)Re(
)(
3 zzw
wh
zy
zx
zfw
Dodecahedron
• Permutation of the Riemann sheets is a symmetric group
• The rotation group of the dodecahedron is isomorphic to the alternating group A5
• A5 is not solvable
Integrated Calculus and Physics at LaGuardia C C
• Frank Y. Wang• fwang@lagcc.cuny.edu• http://faculty.lagcc.cuny.edu/fwang
• http://www.wiley.com