A new identity of Dickson polynomials

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 Müller--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.

In this version, a few minor errors are fixed, some proofs are simplified, and the last two sections are reorganized and shortened

Cited by in corpus (4)

• Solving $x^{2^k+1}+x+a=0$ in ${\mathbb{F}}_{2^n}$ with $\gcd (n,k)=1$
• Solving $X^{q+1}+X+a=0$ over Finite Fields
• Complete solution over $\GF{p^n}$ of the equation $X^{p^k+1}+X+a=0$
• An explanation for the similar appearance of two identities involving Dickson polynomials