Dynamical Behavior and Galois Theory of Iterated Quadratic Polynomials

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Senior Projects Fall 2012

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http://digitalcommons.bard.edu/senproj_f2012/2; http://digitalcommons.bard.edu/cgi/viewcontent.cgi?article=1030&context=senproj_f2012; http://digitalcommons.bard.edu/cgi/viewcontent.cgi?article=1030&context=senproj_f2012
Lazaridis Giannopoul, Damianos
Bard Digital Commons
Iterated Polynomials; Galois Theory; Postcritically Finite Polynomials; Factorization; Dynamical Systems; Number Theory; Set Theory
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The objective of this project is to investigate how polynomials with rational coefficients and their iterates behave modulo p where p is a prime number. The project integrates Number Theory, Galois Theory, Dynamical Systems and Calculus Theory and it makes extensive use of programming. We state conjectures on how an iterated polynomial behaves modulo prime numbers and we explain most of this behavior by examining the role of the discriminant on the iterates of postcritically finite polynomials and the Galois group of the second iterate of these polynomials.