Abstract
Given a positive integer n, any root of a primitive polynomial x n + a 1 x nāāā1 + ... + a n over the finite field \(\mathbb{F}_q\) of q elements (where q is a power of a prime p) is a primitive (generating) element of the extension \(\mathbb{F}_{q^n}\), by definition having multiplicative order q n ā 1.
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References
Cohen, S.D.: Primitive elements and polynomials with arbitrary trace. Discrete Math.Ā 83, 1ā7 (1990)
Cohen, S.D.: Primitive elements and polynomials: existence results. In: Finite fields, coding theory, and advances in communications and computing. Lecture Notes in Pure and Appl. Math., vol.Ā 141, pp. 43ā55. Dekker, New York (1993)
Cohen, S.D., Mills, D.: Primitive polynomials with first and second coefficients prescribed. Finite Fields Appl.Ā 9, 334ā350 (2003)
Fan, S.-Q., Han, W.-B.: p-adic formal series and primitive polynomials over finite fields. Proc. Amer. Math. Soc.Ā 132, 15ā31 (2004)
Fan, S.-Q., Han, W.-B.: p-adic formal series and Cohenās problem. Glasgow Math. J. (to appear)
Fan, S.-Q., Han, W.-B.: Character sums over Galois rings and primitive polynomials over finite fields. Finite Fields Appl. (to appear)
Han, W.-B.: The coefficients of primitive polynomials over finite fields. Math. Comp.Ā 65, 331ā340 (1996)
Han, W.-B.: On Cohenās problem. In: Chinacrypt 1996, pp. 231ā235. Academic Press, China (1996)
Han, W.-B.: On two exponential sums and their applications. Finite Fields Appl.Ā 3, 115ā130 (1997)
Han, W.-B.: The distribution of the coefficients of primitive polynomials over finite fields. In: Cryptography and computational number theory. Progr. Comput. Sci. Appl. Logic, vol.Ā 20, pp. 43ā57. Birkhuser, Basel (2001)
Jungnickel, D., Vanstone, S.A.: On primitive polynomials over finite fields. J. AlgebraĀ 124, 337ā353 (1989)
Koblitz, N.: p-adic Numbers, p-adic Analysis, and Zeta-Functions. Springer, New York (1984)
Li, W.-C.W.: Character sums over p-adic fields. J. Number TheoryĀ 74, 181ā229 (1999)
Mills, D.: Existence of primitive polynomials with three coefficients prescribed. JP J. Algebra Number Theory Appl. (to appear)
Ren, D.-B.: On the coefficients of primitive polynomials over finite fields. Sichuan Daxue XuebaoĀ 38, 33ā36
Shparlinski, I.E.: On primitive polynomials. Prob. Peredachi Inform.Ā 23, 100ā103 (1988) (Russian)
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Cohen, S.D. (2004). Primitive Polynomials over Small Fields. In: Mullen, G.L., Poli, A., Stichtenoth, H. (eds) Finite Fields and Applications. Fq 2003. Lecture Notes in Computer Science, vol 2948. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24633-6_16
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DOI: https://doi.org/10.1007/978-3-540-24633-6_16
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