Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Permutation properties of Dickson and Chebyshev polynomials and
  connections to number theory   The $k$th Dickson polynomial of the first kind, $D_k(x) \in {\mathbb Z}[x]$,
is determined by the formula: $D_k(u+1/u) = u^k + 1/u^k$, where $k \ge 0$ and
$u$ is an indeterminate. These polynomials are closely related to Chebyshev
polynomials and have been widely studied. Leonard Eugene Dickson proved in 1896
that $D_k(x)$ is a permutation polynomial on ${\mathbb F}_{p^n}$, $p$ prime, if
and only if GCD$(k,p^{2n}-1)=1$, and his result easily carries over to
Chebyshev polynomials when $p$ is odd. This article continues on this theme, as
we find special subsets of ${\mathbb F}_{p^n}$ that are stabilized or permuted
by Dickson or Chebyshev polynomials. Our analysis also leads to a factorization
formula for Dickson and Chebyshev polynomials and some new results in
elementary number theory. For example, we show that if $q$ is an odd prime
power, then $\prod\left\{ a \in{\mathbb F}_q^\times : \text{$a$ and $4-a$ are
nonsquares} \right\} = 2$.

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The categories of this paper are: chao-dyn
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The categories of this paper are: solv-int
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The categories of this paper are: ao-sci
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The categories of this paper are: bayes-an
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The categories of this paper are: cs
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The categories of this paper are: alg-geom
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The categories of this paper are: q-bio
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The categories of this paper are: hep-th
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The categories of this paper are: adap-org
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: acc-phys
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The categories of this paper are: funct-an
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: supr-con