Abstract
Let F q be a finite field of characteristic p. In this paper, by using the index sum method the authors obtain a sufficient condition for the existence of a primitive element \(\alpha \in {F_{{q^n}}}\) such that α + α −1 is also primitive or α + α −1 is primitive and α is a normal element of \({F_{{q^n}}}\) over F q.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Carlitz, L., Primitive roots in finite fields, Trans. Am. Math. Soc., 73, 1952, 373–382.
Carlitz, L., Some problems involving primitive roots in a finite field, Proc. Nat. Acad. Sci. U. S. A., 38, 1952, 314–318.
Chou, W. S. and Cohen, S. D., Primitive elements with zero traces, Finite Fields Appl., 7, 2001, 125–141.
Cohen, S. D., Primitive elements and polynomials: Existence results, Lecture Notes in Pure and Appl., http:// wwwresearchgatenet/publication/265461791
Cohen, S. D., Primitive roots in the quadratic extension of a finite field, J. London Math Soc., 27(2), 1983, 221–228.
Cohen, S. D., Primitive elements and polynomials with arbitrary trace, Discrete Math., 83, 1990, 1–7.
Cohen, S. D. and Hachenberger, D., Primitive normal bases with prescribed trace, Applicable Algebra in Engineering Communication and Computing, 9, 1999, 383–403.
Cohen, S. D. and Huczynska, S., Primitive free quartics with specified norm and trace, Acta Arith., 109(4), 2003, 359–385.
Dahab, R., Hankerson, D., Hu, F., et al., Software multiplication using Gaussian normal bases, IEEE Trans. Comput., 55, 2006, 974–984.
Davenport, H., Bases for finite fields, J. London Math. Soc., 43, 1968, 21–39.
Fan, S. Q., Han, W. B., Feng, K. Q. and Zhang, X. Y., Primitive normal polynomials with the first two coefficients prescribed: A revised p-adic method, Finite Fields Appl., 13(3), 2007, 577–604.
Fan, S. Q., Han, W. B. and Feng, K. Q., Primitive normal polynomials with multiple coefficients prescribed: An asymptotic result, Finite Fields Appl., 13(4), 2007, 1029–1044.
Fan, S. Q., Primitive normal polynomials with the last half coefficients prescribed, Finite Fields Appl., 15(5), 2009, 604–614.
He, L. B. and Han, W. B., Research on primitive elements in the form a + a-1 over Fq, Journal of Information Engineering University, 4(2), 2003, 97–98 (in Chinese).
Lenstra, H. W. and Schoof, R. J., Primitive normal bases for finite fields, Math. Comp., 48, 1987, 271–231.
Lidl, R. and Niederreiter, H., Finite Fields, Cambridge University Press, Cambridge, 1987.
Lidl, R. and Niederreiter, H., Finite Fields and Their Applications, 2nd edition, Cambrige University Press, Cambrige, 1994.
Morgan, I. H. and Mullen, L. G., Primitive normal polynomials over finite fields, Math. Comp., 63, 1994, 759–765.
Qi, W. F. and Tian, T., Primitive normal element and its inverse in finite fields, Acta Mathematics Sinica (Chinese Series), 49(3), 2006, 657–668.
Wan, D. Q., Generators and irreducible polynomials over finite fields, Mathematics of Computation, 66, 1997, 1195–1212.
Wang, P. P., Cao, X. W. and Feng, R. Q., On the existence of some specific elements in finite fields of characteristic 2, Finite Fields Appl., 18(4), 2012, 800–813.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author is supported by the National Natural Science Foundation of China (No. 11401408), the Natural Science Foundation of Sichuan Province (No. 14ZA0034) and the Sichuan Normal University Key Project Foundation (No. 13ZDL06). The second author is supported by the National Natural Science Foundation of China (No. 11001170) and the Natural Science Foundation of Shanghai Municipal (No. 13ZR1422500).
Rights and permissions
About this article
Cite this article
Liao, Q., Li, J. & Pu, K. On the existence for some special primitive elements in finite fields. Chin. Ann. Math. Ser. B 37, 259–266 (2016). https://doi.org/10.1007/s11401-016-0949-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-016-0949-5