Research article

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

  • 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.

    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

    Related Papers:

  • 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.


    加载中


    [1] L. Carlitz, A theorem of Dickson on irreducible polynomials, P. Am. Math. Soc., 3 (1952), 693-700. doi: 10.1090/S0002-9939-1952-0049940-6
    [2] K. M. Cheng, Permutational behavior of reversed Dickson polynomials over finite fields II, AIMS. Math., 2 (2017), 586-609. doi: 10.3934/Math.2017.4.586
    [3] K. M. Cheng, S. F. Hong, The first and second moments of reversed Dickson polynomials over finite fields, J. Number Theory, 187 (2018), 166-188. doi: 10.1016/j.jnt.2017.10.021
    [4] C. F. Gauss, Arithmetische Untersuchungen, Chelsea, 1965.
    [5] H. Huang, S. M. Han, W. Cao, Normal bases and irreducible polynomials, Finite Fields Th. App., 50 (2018), 272-278. doi: 10.1016/j.ffa.2017.12.004
    [6] R. Lidl, H. Niederreiter, Finite Fields, Cambridge University Press, 1997.
    [7] G. L. Mullen, D. Panario, Handbook of Finite Fields, CRC Press, 2013.
    [8] O. Ore, Theory of non-commutative polynomials, Ann. Math., 34 (1933), 480-508. doi: 10.2307/1968173
    [9] O. Ore, On a special class of polynomials, T. Am. Math. Soc., 35 (1933), 559-584. doi: 10.1090/S0002-9947-1933-1501703-0
    [10] O. Ore, Contributions to the theory of finite fields, T. Am. Math. Soc., 36 (1934), 243-274. doi: 10.1090/S0002-9947-1934-1501740-7
    [11] O. Ore, Some studies on cyclic determinants, Duke Math. J., 18 (1951), 343-354. doi: 10.1215/S0012-7094-51-01825-X
    [12] F. Ruskey, C. R. Miers, J. Sawada, The number of irreducible polynomials and Lyndon words with given trace, SIAM J. Discrete Math., 14 (2001), 240-245. doi: 10.1137/S0895480100368050
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3502) PDF downloads(333) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog