Published online by Cambridge University Press: 20 November 2018
Fix an integer $m>1$, and set ${{\zeta }_{m}}=\exp \left( 2\pi i/m \right)$. Let $\bar{x}$ denote the multiplicative inverse of $x$ modulo $m$. The Kloosterman sums $R\left( d \right)=\sum\limits_{x}{\zeta _{m}^{x+d\bar{x}}},1\le d\le m,\left( d,m \right)=1$, satisfy the polynomial
where the sum and product are taken over a complete system of reduced residues modulo $m$. Here we give a natural factorization of ${{f}_{m}}\left( x \right)$, namely,
where $\sigma$ runs through the square classes of the group $Z_{m}^{*}$ of reduced residues modulo $m$. Questions concerning the explicit determination of the factors $f_{m}^{\left( \sigma \right)}\left( x \right)$ (or at least their beginning coefficients), their reducibility over the rational field $\text{Q}$ and duplication among the factors are studied. The treatment is similar to what has been done for period polynomials for finite fields.