In this paper, we present some necessary and sufficient conditions under which an irreducible polynomial is self-reciprocal (SR) or self-conjugate-reciprocal (SCR). By these characterizations, we obtain some enumeration formulas of SR and SCR irreducible factors of $x^n-\lambda$, $\lambda \in {\mathbb{F}}_q^{⁎}$, over ${\mathbb{F}}_q$, which are just open questions posed by Boripan et al. (2019). We also count the numbers of Euclidean and Hermitian LCD constacyclic codes and show some well-known results on Euclidean and Hermitian self-dual constacyclic codes in a simple and direct way.

中文翻译：

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x ⁿ  -  λ的自反和自共轭倒数不可约因子及其应用

在本文中，我们提出了一个不可约多项式是自可逆（SR）或自共轭可逆（SCR）的必要条件和充分条件。通过这些表征，我们得到了SR和SCR不可约因子的一些计算公式。$X^{ñ}-\lambda$， $\lambda \in {\mathbb{F}}_q^{⁎}$，结束 ${\mathbb{F}}_q$，这只是Boripan等人提出的开放性问题。（2019）。我们还计算了欧几里得和厄米的LCD固定代码的数目，并以简单直接的方式展示了一些关于欧几里得和厄米的自对偶固定代码的结果。