Vol. 16, No. 4, 2022

Download this article
Download this article For screen
For printing
Recent Issues

Volume 18, 1 issue

Volume 17, 12 issues

Volume 16, 10 issues

Volume 15, 10 issues

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
Other MSP Journals
Parametrizing roots of polynomial congruences

Matthew Welsh

Vol. 16 (2022), No. 4, 881–918

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).

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