A new identity of Dickson polynomials

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.