In this paper, we present two constructions of irreducible (primitive) polynomials over ${\mathbb{F}}_q$ of degree rm from irreducible (primitive) polynomial over ${\mathbb{F}}_{q^m}$ of degree r, and we show that these two constructions coincide. The first construction is based on the Frobenius automorphism of ${\mathbb{F}}_{q^m}$ over ${\mathbb{F}}_q$. The second one comes from a generalization of a construction of primitive polynomials over ${\mathbb{F}}_q$ which uses the companion matrix. From this generalization, given an irreducible (resp. primitive) polynomial over ${\mathbb{F}}_{q^m}$ of degree r, we generate multiple (resp. all) irreducible polynomials over ${\mathbb{F}}_q$ of degree rm. As an application, a characterization of the generator polynomial of a BCH code over ${\mathbb{F}}_q$ is given. Then, we show how two BCH codes over ${\mathbb{F}}_q$ and ${\mathbb{F}}_{q^m}$, respectively, and their generator polynomials are related.

在本文中，我们提出了两个不可约（原始）多项式的构造 ${\mathbb{F}}_q$度RM从束缚（原语）多项式过${\mathbb{F}}_{q^{米}}$r度，我们证明这两个结构是一致的。第一个构造基于 Frobenius 自同构${\mathbb{F}}_{q^{米}}$ 超过 ${\mathbb{F}}_q$. 第二个来自对本原多项式构造的推广${\mathbb{F}}_q$它使用伴随矩阵。从这个概括，给定一个不可约的（分别是原始的）多项式${\mathbb{F}}_{q^{米}}$度[R ，我们生成多个（相应地，所有）不可约多项式过${\mathbb{F}}_q$程度rm。作为一个应用，BCH 码生成多项式的表征${\mathbb{F}}_q$给出。然后，我们展示了两个 BCH 代码如何通过${\mathbb{F}}_q$ 和 ${\mathbb{F}}_{q^{米}}$，分别与它们的生成多项式相关。