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 μ(modm) of the polynomial congruence F(μ) 0(modm), F(X) [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) = X3 2. We show that only a special class of ideals are needed to parametrize the roots μ(modm), and that in the cubic setting, d = 3, general ideals correspond to pairs of roots μ1(modm1), μ2(modm2) satisfying gcd (m1,m2,μ1 μ2) = 1. At the end we illustrate our parametrization and this correspondence between roots and ideals with a few applications, including finding approximations to μ m , finding an explicit Euler product for the cotype zeta function of [213], and computing the composition of cubic ideals in terms of the roots μ1(modm1) and μ2(modm2).

Keywords
roots of congruences, parametrization, polynomial congruences, cubic congruences, ideals
Mathematical Subject Classification
Primary: 11A07
Secondary: 11C08, 11F99, 11R47
Milestones
Received: 5 August 2020
Revised: 6 July 2021
Accepted: 5 August 2021
Published: 5 August 2022
Authors
Matthew Welsh
School of Mathematics
University of Bristol
Bristol
United Kingdom