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

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.