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2022 Parametrizing roots of polynomial congruences
Matthew Welsh
Algebra Number Theory 16(4): 881-918 (2022). DOI: 10.2140/ant.2022.16.881

## Abstract

We use the arithmetic of ideals in orders to parametrize the roots $\mu\hspace{0.3em}(\mod\hspace{0.3em}m)$ of the polynomial congruence $F(\mu)\equiv0\hspace{0.3em}(\mod\hspace{0.3em}m)$, $F(X)\in\mathbb{Z}\lbrack X\rbrack$ 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 $\mu\hspace{0.3em}(\mod\hspace{0.3em}m)$, and that in the cubic setting, $d=3$, general ideals correspond to pairs of roots $\mu_1\hspace{0.3em}(\mod\hspace{0.3em}m_1)$, $\mu_2\hspace{0.3em}(\mod\hspace{0.3em}m_2)$ 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 $\frac\mu m\in\mathbb{R}/\mathbb{Z}$, finding an explicit Euler product for the cotype zeta function of $\mathbb{Z}\lbrack2^{1/3}\rbrack$, and computing the composition of cubic ideals in terms of the roots $\mu_1\hspace{0.3em}(\mod\hspace{0.3em}m_1)$ and $\mu_2\hspace{0.3em}(\mod\hspace{0.3em}m_2)$.

## Citation

Matthew Welsh. "Parametrizing roots of polynomial congruences." Algebra Number Theory 16 (4) 881 - 918, 2022. https://doi.org/10.2140/ant.2022.16.881

## Information

Received: 5 August 2020; Revised: 6 July 2021; Accepted: 5 August 2021; Published: 2022
First available in Project Euclid: 18 August 2022

Digital Object Identifier: 10.2140/ant.2022.16.881

Subjects:
Primary: 11A07
Secondary: 11C08 , 11F99 , 11R47

Keywords: cubic congruences , Ideals , parametrization , polynomial congruences , roots of congruences

Rights: Copyright © 2022 Mathematical Sciences Publishers

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