Let \(n,k\in \mathbb N\), \(p\) is odd prime, \(p>n\), and consider a polynomial with integer coefficients \(f(x)=a_n x^n+ \ldots +a_1 x\) in the case \(\gcd (a_n,\ldots,a_1,p)=1\). Sums of the form
\[ S(p^k) =\sum_{x=1}^q e^{2\pi if(x)/q}, \]
where \(q=p^k\), are called complete rational exponential sums. A. Weil proved that \(S(p)\leq (n-1) \sqrt{p}\).
The present paper studies \(S(p^k)\) and obtains a similar result in the case \(k>1\) and the congruence \(f'(x)\equiv 0 \pmod p\), \(0 \leq x < p\) has no multiple roots.