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

Weihua Li; Chengcheng Fang; Wei Cao; Weihua Li; Chengcheng Fang; Wei Cao

doi:10.3934/math.2020185

AIMS Mathematics

2020, Volume 5, Issue 4: 2877-2887. doi: 10.3934/math.2020185

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Research article

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

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

Received: 04 December 2019 Accepted: 03 February 2020 Published: 18 March 2020
MSC : 11T06, 11T55

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.
- finite field,
- irreducible polynomial,
- linearized polynomial
Citation: Weihua Li, Chengcheng Fang, Wei Cao. On the number of irreducible polynomials of special kinds in finite fields[J]. AIMS Mathematics, 2020, 5(4): 2877-2887. doi: 10.3934/math.2020185

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Abstract

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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