#### Vol. 16, No. 4, 2022

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Parametrizing roots of polynomial congruences

### Matthew Welsh

Vol. 16 (2022), No. 4, 881–918
##### Abstract

We use the arithmetic of ideals in orders to parametrize the roots $\mu \phantom{\rule{0.3em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.3em}{0ex}}m\right)$ of the polynomial congruence $F\left(\mu \right)\equiv 0\phantom{\rule{0.3em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.3em}{0ex}}m\right)$, $F\left(X\right)\in ℤ\left[X\right]$ 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 $F\left(X\right)={X}^{3}-2$. We show that only a special class of ideals are needed to parametrize the roots $\mu \phantom{\rule{0.3em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.3em}{0ex}}m\right)$, and that in the cubic setting, $d=3$, general ideals correspond to pairs of roots ${\mu }_{1}\phantom{\rule{0.3em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.3em}{0ex}}{m}_{1}\right)$, ${\mu }_{2}\phantom{\rule{0.3em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.3em}{0ex}}{m}_{2}\right)$ satisfying $\mathrm{gcd}\left({m}_{1},{m}_{2},{\mu }_{1}-{\mu }_{2}\right)=1$. At the end we illustrate our parametrization and this correspondence between roots and ideals with a few applications, including finding approximations to $\frac{\mu }{m}\in ℝ∕ℤ$, finding an explicit Euler product for the cotype zeta function of $ℤ\left[{2}^{1∕3}\right]$, and computing the composition of cubic ideals in terms of the roots ${\mu }_{1}\phantom{\rule{0.3em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.3em}{0ex}}{m}_{1}\right)$ and ${\mu }_{2}\phantom{\rule{0.3em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.3em}{0ex}}{m}_{2}\right)$.

##### Keywords
roots of congruences, parametrization, polynomial congruences, cubic congruences, ideals
##### Mathematical Subject Classification
Primary: 11A07
Secondary: 11C08, 11F99, 11R47