Roots of certain polynomials over finite fields.

We determine the roots in F q 3 of the polynomial X 2 q ℓ + 1 + X + a for each positive integer ℓ and each a ∈ F q , where q is a power of 2. We introduce a new approach for this type of question, and we obtain results which are more explicit than the previous results in this area. Our results resolve an open problem and a conjecture of Zheng, Kan, Zhang, Peng, and Li. [ABSTRACT FROM AUTHOR]