= Power Gauss sum over a prime field
{title2=$G_{a,p}=p^{-1}\sum_{x\in\mathbb F_p}e^{2\pi iax^k/p}$}
For $a\ne0$ in the <finite field> $\mathbb F_p$, put $d=\gcd(k,p-1)$. The <multiplicative group of a finite field is cyclic>, and <character orthogonality> expresses the number of $k$th roots of $y\ne0$ as $\sum_{\chi^d=1}\chi(y)$. The trivial <character> contributes $-1$, cancelling the $x=0$ term. Each of the remaining $d-1$ <Gauss sums of Dirichlet characters> has modulus $\sqrt p$, by the <prime-field character Gauss sum identity>. Consequently $|G_{a,p}|\le(d-1)/\sqrt p\le k/\sqrt p$.
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