Parametrizing roots of polynomial congruences

Abstract

We use the arithmetic of ideals in orders to parametrize the roots $\mu (\mathrm{mod}m)$ of the polynomial congruence $F(\mu )\equiv 0(\mathrm{mod}m)$, $F(X)\in \mathbb{Z}[X]$ monic, irreducible and degree $d$. Our parametrization generalizes Gauss’s classic parametrization of the roots of quadratic congruences using binary quadratic forms, which had previously only been extended to the cubic polynomial $F(X)=X^3-2$. We show that only a special class of ideals are needed to parametrize the roots $\mu (\mathrm{mod}m)$, and that in the cubic setting, $d=3$, general ideals correspond to pairs of roots ${\mu }_1(\mathrm{mod}{m}_1)$, ${\mu }_2(\mathrm{mod}{m}_2)$ satisfying $\gcd (m_1,m_2,{\mu }_1–{\mu }_2)=1$. At the end we illustrate our parametrization and this correspondence between roots and ideals with a few applications, including finding approximations to $\frac{\mu }{m}\in ℝ∕\mathbb{Z}$, finding an explicit Euler product for the cotype zeta function of $\mathbb{Z}[2^{1∕3}]$, and computing the composition of cubic ideals in terms of the roots ${\mu }_1(\mathrm{mod}{m}_1)$ and ${\mu }_2(\mathrm{mod}{m}_2)$.