We use the arithmetic of ideals in orders to parametrize the roots $\mu (\mathrm{mod}<!--FUNCTION\; APPLICATION-->m)$ of the polynomial congruence $F(\mu )\equiv 0(\mathrm{mod}<!--FUNCTION\; APPLICATION-->m)$, $F(X)\in \mathbb{Z}[X]$ 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 $\mu (\mathrm{mod}<!--FUNCTION\; APPLICATION-->m)$, and that in the cubic setting, $d=3$, general ideals correspond to pairs of roots ${\mu }_1(\mathrm{mod}<!--FUNCTION\; APPLICATION-->m_1)$, ${\mu }_2(\mathrm{mod}<!--FUNCTION\; APPLICATION-->m_2)$ satisfying $\gcd <!--FUNCTION\; APPLICATION-->(m_1,m_2,{\mu }_1-{\mu }_2)=1$. 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 ${\mu }_1(\mathrm{mod}<!--FUNCTION\; APPLICATION-->m_1)$ and ${\mu }_2(\mathrm{mod}<!--FUNCTION\; APPLICATION-->m_2)$.

我们使用理想的算术来参数化根$\mu (\mathrm{模组}<!--FUNCTION\; APPLICATION-->米)$多项式同余的$F(\mu )\equiv 0(\mathrm{模组}<!--FUNCTION\; APPLICATION-->米)$,$F(X)\in \mathbb{Z}[X]$monic、不可约和度数$d$. 我们的参数化使用二元二次形式推广了 Gauss 对二次同余根的经典参数化，以前只扩展到三次多项式$F(X)=X^3-2$. 我们证明了只需要一类特殊的理想来参数化根$\mu (\mathrm{模组}<!--FUNCTION\; APPLICATION-->米)$，并且在立方设置中，$d=3$, 一般理想对应于成对的根${\mu }_1(\mathrm{模组}<!--FUNCTION\; APPLICATION-->{米}_1)$,${\mu }_2(\mathrm{模组}<!--FUNCTION\; APPLICATION-->{米}_2)$令人满意的$\gcd <!--FUNCTION\; APPLICATION-->({米}_1,{米}_2,{\mu }_1-{\mu }_2)=1$. 最后，我们通过一些应用来说明我们的参数化以及根和理想之间的这种对应关系，包括找到$\frac{\mu }{米}\in ℝ∕\mathbb{Z}$, 为 cotype zeta 函数找到一个显式的欧拉积$\mathbb{Z}[2^{1∕3}]$，并根据根计算三次理想的组成${\mu }_1(\mathrm{模组}<!--FUNCTION\; APPLICATION-->{米}_1)$和${\mu }_2(\mathrm{模组}<!--FUNCTION\; APPLICATION-->{米}_2)$.