Primitive normal bases for finite fields

Authors:
H. W. Lenstra and R. J. Schoof

Journal:
Math. Comp. **48** (1987), 217-231

MSC:
Primary 11T30; Secondary 12E20

DOI:
https://doi.org/10.1090/S0025-5718-1987-0866111-3

MathSciNet review:
866111

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Abstract | References | Similar Articles | Additional Information

Abstract: It is proved that any finite extension of a finite field has a normal basis consisting of primitive roots.

**[1]**Jacob T. B. Beard Jr. and Karen I. West,*Some primitive polynomials of the third kind*, Math. Comp.**28**(1974), 1166–1167; addendum, ibid. 28 (1974), no. 128, loose microfiche suppl. C1–C5. MR**0366879**, https://doi.org/10.1090/S0025-5718-1974-0366879-X**[2]**L. Carlitz,*Primitive roots in a finite field*, Trans. Amer. Math. Soc.**73**(1952), 373–382. MR**0051869**, https://doi.org/10.1090/S0002-9947-1952-0051869-9**[3]**L. Carlitz,*Some problems involving primitive roots in a finite field*, Proc. Nat. Acad. Sci. U.S.A.**38**(1952), 314–318; errata, 618. MR**0049939****[4]**H. Davenport,*Bases for finite fields*, J. London Math. Soc.**43**(1968), 21–39. MR**0227144**, https://doi.org/10.1112/jlms/s1-43.1.21**[5]**G. H. Hardy & E. M. Wright,*An Introduction to the Theory of Numbers*, 4th ed., Oxford University Press, Oxford, 1968.**[6]**W. H. Mills,*The degrees of the factors of certain polynomials over finite fields.*, Proc. Amer. Math. Soc.**25**(1970), 860–863. MR**0263783**, https://doi.org/10.1090/S0002-9939-1970-0263783-0**[7]**Oystein Ore,*Contributions to the theory of finite fields*, Trans. Amer. Math. Soc.**36**(1934), no. 2, 243–274. MR**1501740**, https://doi.org/10.1090/S0002-9947-1934-1501740-7**[8]**Jean-Pierre Serre,*Cours d’arithmétique*, Collection SUP: “Le Mathématicien”, vol. 2, Presses Universitaires de France, Paris, 1970 (French). MR**0255476****[9]**Neal Zierler,*On the theorem of Gleason and Marsh*, Proc. Amer. Math. Soc.**9**(1958), 236–237. MR**0094332**, https://doi.org/10.1090/S0002-9939-1958-0094332-2**[10]**Neal Zierler,*On 𝑥ⁿ+𝑥+1 over 𝐺𝐹(2)*, Information and Control**16**(1970), 502–505. MR**0271072**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1987-0866111-3

Keywords:
Finite field,
normal basis,
primitive root

Article copyright:
© Copyright 1987
American Mathematical Society