Chebotarev定理与递归线性方程组的同余关系及$\mathbb{F}_p$上的因式分解算法
Théorème de Chebotarev et Congruences de suites récurrentes linéaires, liens avec les algorithmes de factorisations sur $\mathbb{F}_p$
摘要：
Fibonacci和Lucas序列所满足的经典同余关系反映在由黄金数生成的环中素数的分解上。这推广到建立一个对应关系，我们希望将是新的Chebotarev定理和线性序列满足的同余。这个链接是在数字域扩张的上下文中完成的。特别地，我们刻画了由线性递归序列项上的单同余完全分解的素理想。我们的结果被大量的例子所证明，包括$S_3$群的Padovan序列和与$\mathbb{PSL}_2 (\mathbb{F}_7)$群的Cartier Trink多项式相关的Padovan序列.此外，我们建立了一个链接之间的因式分解算法的Berlekamp和Cantor-Zassenhaus和这项工作的结果。
英文摘要：
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.
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