SOLUBILITY TO THE LEGENDRE EQUATION IN THE POLYNOMIAL RING $\mathbb{F}_q[t]$
Keywords:
Legendre equation, polynomial rings, finite fieldsDOI:
https://doi.org/10.17654/0972555525038Abstract
We determine the necessary and sufficient conditions for the existence of nontrivial solutions to the Legendre equation $a x^2+b y^2+c z^2=0$ over the ring of polynomials $\mathbb{F}_q[t]$, where $\mathbb{F}_q$ is a finite field. Additionally, we present a method for determining whether the Legendre equation has nontrivial solutions in $\mathbb{F}_q$. When such a solution exists, we can construct an infinite number of solutions in $\mathbb{F}_q[t]$ via the Réalis formulae.
Received: August 23, 2025
Revised: September 17, 2025
Accepted: September 25, 2025
References
[1] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system, I. The user language, J. Symbolic Comput. 24 (1997), 235-265.
http://dx.doi.org/10.1006/jsco.1996.0125.
Computational Algebra and Number Theory (London, 1993).
[2] H. Chevalley and E. Warning, Bemerkung zur vorstehenden Arbeit, Abhandlungen Aus Dem Mathematischen Seminar Der Universität Hamburg, 11 (1935), pp. 76-83.
[3] L. E. Dickson, History of the Theory of Numbers, Volume II: Diophantine Analysis, Carnegie Institution of Washington, 1920, pp. 422-423.
[4] P. G. L. Dirichlet and R. Dedekind, Vorlesungen über zahlentheorie, Art. (1871), 156-157.
[5] O. Hemer, On the solvability of the diophantine equation in imaginary Euclidean quadratic fields, Arkiv Für Matematik 2 (1952), 57-82.
[6] J. Leal-Ruperto, La ecuación de Legendre en los Enteros de Gauss y en el Anillo de los Polinomios Racionales, Tesis Doctoral (UMA Editorial, 2015).
[7] M. Legendre, Théorème sur la possibilité des équations indeterminées du second degré, Histoire de l’Academie Royale des Sciences (1785), 507-513.
[8] R. Lidl and H. Niederreiter, Finite fields, Cambridge University Press, 1997.
[9] T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, pp. 218-222.
[10] P. Samet, An equation in Gaussian integers, The American Mathematical Monthly 59 (1952), 448-452.
Downloads
Published
Issue
Section
License
Copyright (c) 2025 PUSHPA PUBLISHING HOUSE, PRAYAGRAJ, INDIA
This work is licensed under a Creative Commons Attribution 4.0 International License.
_________________________________
Attribution: Credit Pusha Publishing House as the original publisher, including title and author(s) if applicable.
Non-Commercial Use: For non-commercial purposes only. No commercial activities without explicit permission.
No Derivatives: Modifying or creating derivative works not allowed without written permission.
Contact Pusha Publishing House for more info or permissions.
