AIMS Mathematics, 2020, 5(4): 2877-2887. doi: 10.3934/math.2020185

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On the number of irreducible polynomials of special kinds in finite fields

School of Mathematics and Statistics, Ningbo University, Ningbo 315211, P. R. China

Let $\mathbb{F}_q$ be the finite field of order $q$ and $f(x)$ be an irreducible polynomial of degree $n$ over $\mathbb{F} _q$. For a positive divisor $n_1$ of $n$, define the $n_1$-traces of $f(x)$ to be $\mathrm{Tr}(\alpha;n_1)=\alpha+\alpha^q+\cdots+\alpha^{q^{n_1-1}}$ where $\alpha$'s are the roots of $f(x)$. Let $N_q^*(n;n_1)$ denote the number of monic irreducible polynomials of degree $n$ over $\mathbb{F} _q$ with nozero $n_1$-traces. Ruskey, Miers and Sawada have found the formula for $N_q^*(n;n)$. Based on the properties of linearized polynomials, we obtain the formula for $N_q^*(n;n_1)$ in the general case, including a new proof to the result by Ruskey, Miers and Sawada.
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