Some arithmetic properties of the minimal polynomials of Gauss sums

Abstract:

For the minimal polynomial $f(x) = {x^k} + {b_1}{x^{k - 1}} + \cdots + {b_k}$ of $\sum \nolimits _{n = 0}^{p - 1} {\exp (2\pi i{n^k}/p)}$ over $Q$, where $p$ is a $\operatorname {prime} \equiv 1(\bmod k)$, we evaluate ${b_1},{b_2}$ and prove $\left . p \right |{b_i}(i = 1, \ldots ,k)$ but ${p^2}\nmid {b_j}(j = 2,k)$. Also, we raise the interesting conjecture that ${p^2}\nmid {b_j}$ for $k \geq j \geq 2$.