Hostname: page-component-5cf477f64f-zrtmk Total loading time: 0 Render date: 2025-04-01T17:44:48.122Z Has data issue: false hasContentIssue false
  • English
  • Français

GENERATORS OF FINITE FIELDS WITH PRESCRIBED TRACES

Part of: Finite fields and commutative rings (number-theoretic aspects) General field theory

Published online by Cambridge University Press:  27 May 2021

LUCAS REIS  and
SÁVIO RIBAS
LUCAS REIS
Affiliation:
Departamento de Matemática, Universidade Federal de Minas Gerais (UFMG), Belo Horizonte, MG30270-901, Brazil
SÁVIO RIBAS
Affiliation:
Departamento de Matemática, Universidade Federal de Ouro Preto (UFOP), Ouro Preto, MG35400-000, Brazil e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper explores the existence and distribution of primitive elements in finite field extensions with prescribed traces in several intermediate field extensions. Our main result provides an inequality-like condition to ensure the existence of such elements. We then derive concrete existence results for a special class of intermediate extensions.

Keywords

character sumsfield tracefinite fieldsprimitive elements

MSC classification

Primary: 12E20: Finite fields (field-theoretic aspects)
Secondary: 11T24: Other character sums and Gauss sums
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 112 , Issue 3 , June 2022 , pp. 355 - 366
Check for updates
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Communicated by Dzmitry Badziahin

References

Cohen, S. D., ‘Primitive elements and polynomials with arbitrary trace’, Discrete Math. 83(1) (1990), 1–7.CrossRefGoogle Scholar
Cohen, S. D. and Huczynska, S., ‘The primitive normal basis theorem—without a computer’, J. Lond. Math. Soc. 67(1) (2003), 41–56.CrossRefGoogle Scholar
Cohen, S. D., Oliveira e Silva, T. and Trudgian, T., ‘On consecutive primitive elements in a finite field’, Bull. Lond. Math. Soc. 47(3) (2015), 418–426.CrossRefGoogle Scholar
Diffie, W. and Hellman, M., ‘New directions in cryptography’, IEEE Trans. Inform. Theory 22(6) (1976), 644–654.CrossRefGoogle Scholar
Kapetanakis, G. and Reis, L., ‘Variations of the primitive normal basis theorem’, Des. Codes Cryptogr. 87(7) (2019), 1459–1480.CrossRefGoogle Scholar
Lenstra, H. W. Jr and Schoof, R. J., ‘Primitive normal bases for finite fields’, Math. Comp. 48(177) (1987), 217–231.CrossRefGoogle Scholar
Reis, L., ‘Counting solutions of special linear equations over finite fields’, Finite Fields Appl. 68 (2020), 101759.CrossRefGoogle Scholar