Chebyshev action on finite fields

Given a polynomial ϕ(x) and a finite field Fq one can construct a directed graph where the vertices are the values in the finite field, and emanating from each vertex is an edge joining the vertex to its image under ϕ. When ϕ is a Chebyshev polynomial of prime degree, the graphs display an unusual d...

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Bibliographic Details
Published in:Discrete mathematics 2014-02, Vol.315-316, p.83-94
Main Author: Alden Gassert, T.
Format: Article
Language:English
Subjects:
Chebyshev approximation
Chebyshev polynomial
Emission
Finite field
Graph theory
Graphs
Iterated polynomial
Mathematical analysis
Polynomials
Post-critically finite map
Prime decomposition
Radicals
Symmetry
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Summary:Given a polynomial ϕ(x) and a finite field Fq one can construct a directed graph where the vertices are the values in the finite field, and emanating from each vertex is an edge joining the vertex to its image under ϕ. When ϕ is a Chebyshev polynomial of prime degree, the graphs display an unusual degree of symmetry. In this paper we provide a complete description of these graphs, and then use these graphs to determine the decomposition of primes in the Chebyshev radical extensions.
ISSN:0012-365X
1872-681X
DOI:10.1016/j.disc.2013.10.014