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

## Abstract    Full Text(HTML)    Figure/Table    Related pages

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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# References

1. L. Carlitz, A theorem of Dickson on irreducible polynomials, P. Am. Math. Soc., 3 (1952), 693-700.

2. K. M. Cheng, Permutational behavior of reversed Dickson polynomials over finite fields II, AIMS. Math., 2 (2017), 586-609.

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.

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.

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.

9. O. Ore, On a special class of polynomials, T. Am. Math. Soc., 35 (1933), 559-584.

10. O. Ore, Contributions to the theory of finite fields, T. Am. Math. Soc., 36 (1934), 243-274.

11. O. Ore, Some studies on cyclic determinants, Duke Math. J., 18 (1951), 343-354.

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.