Note on the Kloosterman sum

Abstract:

The Kloosterman sum \[ \sum \limits _{x = 0;(x,p) = 1}^{{p^\alpha } - 1} {\exp (2\pi in(x + \bar x)/{p^\alpha }),} \] where p is an odd prime, $\alpha \geqq 2$ and $(n,p) = 1$, is evaluated in a very short direct way.