Permutation polynomials have many applications in finite fields theory, coding theory, cryptography, combinatorial design, communication theory, and so on. Permutation binomials of the form $x^{ r}(x^{ q-1}+a)$ over $\mathbb{ F}_{ q^2}$ have been studied before, K. Li, L. Qu and X. Chen proved that they are permutation polynomials if and only if $r=1$ and $a^{ q+1}\not=1$. In this paper, we consider the same binomial, but over finite fields $\mathbb{ F}_{ q^3}$ and $\mathbb{ F}_{ q^e}$. Two different kinds of methods are employed, and some partial results are obtained for them.
Categories: cs.IT math.IT math.NT