Th\'eor\`eme de Chebotarev et Congruences de suites r\'ecurrentes lin\'eaires, liens avec les algorithmes de factorisations sur $\mathbb{F}_p$

TL;DR

This paper explores the relationship between Chebotarev's theorem and congruences of linear recurrent sequences, linking number field theory with factorization algorithms and providing numerous examples.

Contribution

It introduces a novel connection between Chebotarev's theorem and linear sequence congruences within number field extensions, with applications to factorization algorithms.

Findings

Characterization of primes totally decomposed by linear sequence congruences

Illustrations with Padovan sequences and group polynomials

Linking factorization algorithms to number field extension properties

Abstract

The classical congruences satisfied by the Fibonacci and Lucas sequences are reflected with the decomposition of primes in the ring generated by the gold number. This generalizes to establish a correspondence that we hope will be new between Chebotarev's theorem and the congruences satisfied by linear sequences. This link is done into the context of number field extensions. In particular we characterize primes ideals totally decomposed by simple congruences on the terms of linear recurrent sequences. Our results are illustrated by numerous examples, including Padovan sequencesof group $S_3$ and associated with the Cartier Trink polynomial of group $\mathbb{PSL}_2 (\mathbb{F}_7)$. Furthermore, we establish a link between the factorisation algorithms of Berlekamp and Cantor-Zassenhaus and the results of this work.