Fix an integer $m>1$, and set $\zeta_{m}=\exp(2\pi i/m)$. Let ${\bar x}$ denote the multiplicative inverse of $x$ modulo $m$. The Kloosterman sums $R(d)=\sum_{x} \zeta_{m}^{x + d{\bar x}}$, $1 \leq d \leq m$, $(d,m)=1$, satisfy the polynomial $$f_{m}(x) = \prod_{d} (x-R(d)) = x^{\phi(m)} +c_{1} x^{\phi(m)-1} + \dots + c_{\phi(m)},$$ where the sum and product are taken over a complete system of reduced residues modulo $m$. Here we give a natural factorization of $f_{m}(x)$, namely, $$ f_{m}(x) = \prod_{\sigma} f_{m}^{(\sigma)}(x),$$ where $\sigma$ runs through the square classes of the group ${\bf Z}_{m}^{*}$ of reduced residues modulo $m$. Questions concerning the explicit determination of the factors $f_{m}^{(\sigma)}(x)$ (or at least their beginning coefficients), their reducibility over the rational field ${\bf Q}$ and duplication among the factors are studied. The treatment is similar to what has been done for period polynomials for finite fields.

Let $q=p^{r}$ with $p$ an odd prime, and $\mathbf{F}_{q}$ denote the finite field of $q$ elements. Let $\Tr\colon\mathbf{F}_{q} \to\mathbf{F}_{p} $ be the usual trace map and set $\zeta_{p} =\exp(2\pi i/p)$. For any positive integer $e$, define the (modified) Gauss sum $g_{r}(e)$ by $$ g_{r}(e) =\sum_{x\in \mathbf{F}_{q}}\zeta_{p}^{\Tr x^{e}} $$ Recently, Evans gave an elegant determination of $g_{1}(12)$ in terms of $g_{1}(3)$, $g_{1}(4)$ and $g_{1}(6)$ which resolved a sign ambiguity present in a previous evaluation. Here I generalize Evans' result to give a complete determination of the sum $g_{r}(12)$.

Precise, elegant evaluations are given for Gauss sums of orders six and twelve.