A note on evaluations of some exponential sums

1. Introduction. The recent article [1] gives explicit evaluations for exponential sums of the form ${\displaystyle S(a,p^{\alpha }+1):={\sum }_{x{\in }_q}\chi \left(a{x}^{p^{\alpha }+1}\right)}$ where χ is a non-trivial additive character of the finite field ${\displaystyle \_q}$, ${\displaystyle q=p^e}$ odd, and ${\displaystyle a\in {\cdot }_q}$. In my dissertation [5], in particular in [4], I considered more generally the sums S(a,N) for all factors N of ${\displaystyle p^{\alpha }+1}$. The aim of the present note is to evaluate S(a,N) in a short way, following [4]. We note that our result is also valid for even q, and the technique used in our proof can also be used to evaluate certain sums of the form ${\displaystyle {\sum }_{x{\in }_q}\chi (a{x}^{p^{\alpha }+1}+bx)}$.