For this project, we explore nite eld dynamics and the various patterns of cycles of elements that emerge from the manipulation of a function and eld. Given a function f : Fp ! Fp, we can create a directed graph with an edge from c to f(c) for all c 2 Fp. We especially consider polynomials of the form f(x) = xd + c and investigate how varying the values of d and c affect the cycles in a given nite eld, Fp. We analyze data to look for graphs that result in cycles of length p. We also identify functions whose graphs have the same structure. We prove three main theorems. The rst theorem states that for p prime, if d1d2 1(mod p 1), then the functions f1(x) = xd1 + c and f2(x) = xd2 c have isomorphic graphs for any c 2 Zp. The second theorem states that if p 3 (mod 4) and p+1 = 2k then there are exactly p+1 2 choices for c in which the graph of f(x) = xp2 +c has one cycle of length p. The third theorem states that for c 2 Zp if d = p 2 and p + 1 = 4q where q is a prime, then there are p3 2 choices for c in which the graph of f(x) = xd + c has one cycle of length p.
Dr. Jacqueline Anderson, Thesis Advisor
Dr. Irina Seceleanu, Committee Member
Dr. Uma Shama, Committee Member
Copyright and Permissions
Original document was submitted as an Honors Program requirement. Copyright is held by the author.
McCarthy, Catlain. (2019). Finite Field Dynamics: Exploring Isomorphic Graphs and Cycles of Length p. In BSU Honors Program Theses and Projects. Item 388. Available at: https://vc.bridgew.edu/honors_proj/388
Copyright © 2019 Catlain McCarthy