We present the formula for the number of monic irreducible polynomials of degree n over the finite field \({\mathbb {F}}_q\) where the coefficients of \(x^{n-1}\) and x vanish for \(n\ge 3\). In particular, we give a relation between rational points of algebraic curves over finite fields and the number of elements \(a\in {\mathbb {F}}_{q^n}\) for which Trace\((a)=0\) and Trace\((a^{-1})=0\).

中文翻译：

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具有消失迹和倒数迹的有限域上不可约多项式的数目

我们提出了有限域\({\mathbb {F}}_q\)上n次不可约多项式数量的公式，其中\(x^{n-1}\)和x的系数对于\ (n\ge 3\)。特别是，我们给出了有限域上代数曲线的有理点与元素数量\(a\in {\mathbb {F}}_{q^n}\)之间的关系，其中 Trace \((a)= 0\)和跟踪\((a^{-1})=0\)。