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For a Dickson polynomial, we have the following result:

$D_n(x,α)$ is a permutation polynomial for the field with $q$ elements if and only if $n$ is coprime to $q^2−1$.

Suppose I am given an arbitrary polynomial that's not necessarily a Dickson polynomial, say $x^5 + x + 4$. Is there a result similar to the above one that can determine if the polynomial is a permutation polynomial for a field with $q$ elements?

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I assume a permutation polynomial is one that - as a map - is a permutaion of $\mathbb F_q$? –  Hagen von Eitzen Feb 19 at 22:02
    
Hagen von Eitzen - That is correct. A permutation polynomial is a bijection from $\mathbb{F}_q \rightarrow \mathbb{F}_q$. –  Joseph DiNatale Feb 19 at 22:07

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