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

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The most beautiful equation

Julia set c= i

Mandelbrot set and bifurcation Mandelbrot set

Newton’s Method in the Complex Plane

Differential Equations

Isaac Newton Invented Calculus Laws of Motion F=m a Differential Equations

Galileo Galilei Pendulum Motion

Johannes Kepler Planetary Motion

(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-Leffler Dear Sir, … It is a question of integrating …, and I can show that these 3 cases are the only ones [integrable]… Roger Cooke Trans.

Fundamental Quantum Condition

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 ~  e iS[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

Complex Functions

z plane and w plane

z plane and w plane (Riemann’s Hypothesis)

Complex Numbers Cartesian vs Polar Forms

Parametrized curve

Map from z to w

De Moivre’s formula

Winding number (2)

Winding number (3)

Property When a loop is large enough, the winding number is the degree of the polynomial.

Expanding r

Index 1, 2, 3

Index

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 S n, the symmetric group on n symbols. For n>4, S n is not solvable.

Riemann surface

Dodecahedron Permutation of the Riemann sheets is a symmetric group The rotation group of the dodecahedron is isomorphic to the alternating group A 5 A 5 is not solvable

Integrated Calculus and Physics at LaGuardia C C Frank Y. Wang