Let $\mathbb{ F}_{ q}$ be the finite field of characteristic $p$ containing $q = p^{ r}$ elements and $f(x)=ax^{ n} + x^{ m}$ a binomial with coefficients in this field. If some conditions on the gcd of $n-m$ an $q-1$ are satisfied then this polynomial does not permute the elements of the field. We prove in particular that if $f(x) = ax^{ n} + x^{ m}$ permutes $\mathbb{ F}_{ p}$, where $n>m>0$ and $a \in { \mathbb{ F}_{ p}}^{ *}$, then $p -1 \leq (d -1)d$, where $d = { { gcd}}(n-m,p-1)$, and that this bound of $p$ in term of $d$ only, is sharp. We show as well how to obtain in certain cases a permutation binomial over a subfield of $\mathbb{ F}_{ q}$ from a permutation binomial over $\mathbb{ F}_{ q}$.
Categories: math.NT