文章给出有限域~${\mathbb{F}}_{q^2}$上~$x^{q^n+1}-\lambda$~的分解和 首一不可约~$\lambda$-自共轭互反多项式的计数公式, 其中~$q$~是素数方幂, $\lambda \in {\mathbb{F}}_q^{\ast }$. 进一步, 得到了~${\mathbb{F}}_{q^2}$上 ~$x^n+1$~的自共轭互反多项式因子的计数公式. 将此公式应用在负循环码上, ~${\mathbb{F}}_{q^2}$上 厄米特互补对偶负循环码的个数也被确定.

In this paper, we present a factorization of $x^{q^n+1}-\lambda$ over ${\mathbb{F}}_{q^2}$ and the enumeration of self-conjugate-reciprocal monic polynomials over ${\mathbb{F}}_{q^2}$, where $q$ is a prime power and $\lambda \in {\mathbb{F}}_q^{\ast }$. Furthermore, we propose the explicit number of monic self-conjugate-reciprocal irreducible factors of $x^n+1$. As an application to negacyclic codes, the number of linear Hermitian complementary dual negacyclic codes has been evaluated.