On monomial complete permutation polynomials

We investigate monomials $ax^d$ over the finite field with q elements $F_q$, in the case where the degree d is equal to $1+rac{q-1}{q'-1}$ with $q=(q′)^n$ for some n. For n=6 we explicitly list all a's for which $ax^d$ is a complete permutation polynomial (CPP) over $F_q$. Some previous characterization results by Wu et al. for n=4 are also made more explicit by providing a complete list of a's such that $ax^d$ is a CPP. For odd n, we show that if q is large enough with respect to n then $ax^d$ cannot be a CPP over $F_q$, unless q is even, n≡3(mod4), and the trace Tr_{F_q/F_{q′}}(1/a) is equal to 0.