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arXiv · 1610.05853

A new identity of Dickson polynomials

Antonia W. Bluher
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Abstract

A new polynomial identity is found for Dickson polynomials in characteristic 2. The identity is used to prove that the two polynomials $x^{q+1}+x+1/a$ and $C(x)+a$ have the same splitting field over $F$, where $F$ is a field of characteristic 2, $a$ is a nonzero element of $F$, $q=2^n>2$, and $C(x) = x (\sum_{i=0}^{n-1} x^{2^i-1})^{q+1}$ is a M\"uller--Cohen--Matthews polynomial of degree $(q^2-q)/2$. In addition, a new proof is obtained for the known result that $C(x)$ induces a permutation on $F_{2^m}$ if $2m$ and $n$ are relatively prime.

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Antonia W. Bluher. 2016-10-19. A new identity of Dickson polynomials. https://doi.org/10.1016/j.ffa.2022.102012

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