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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.
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Fundamental Theorem of AlgebraA 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
Mandelbrot set and bifurcation • Mandelbrot set
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.
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.
Feynman Sum Over Histories A ~ eiS[g]/ћ Sum over all metrics consistent with given boundary conditions
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)
Property • When a loop is large enough, the winding number is the degree of the polynomial.
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.
Dodecahedron • Permutation of the Riemann sheets is a symmetric group • The rotation group of the dodecahedron is isomorphic to the alternating group A5 • A5is 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
