Primitive Normal Bases with Prescribed Trace

  • S. D. Cohen
  • D. Hachenberger

DOI: 10.1007/s002000050112

Cite this article as:
Cohen, S. & Hachenberger, D. AAECC (1999) 9: 383. doi:10.1007/s002000050112


Let E be a finite degree extension over a finite field F = GF(q), G the Galois group of E over F and let aF be nonzero. We prove the existence of an element w in E satisfying the following conditions:

- w is primitive in E, i.e., w generates the multiplicative group of E (as a module over the ring of integers).

- the set {wggG} of conjugates of w under G forms a normal basis of E over F.

- the (E, F)-trace of w is equal to a.

This result is a strengthening of the primitive normal basis theorem of Lenstra and Schoof [10] and the theorem of Cohen on primitive elements with prescribed trace [3]. It establishes a recent conjecture of Morgan and Mullen [14], who, by means of a computer search, have verified the existence of such elements for the cases in which q≤ 97 and n≤ 6, n being the degree of E over F. Apart from two pairs (F, E) (or (q, n)) we are able to settle the conjecture purely theoretically.

Key words: Finite field, Primitive element, Normal basis, Free element, Trace, Character sum 

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • S. D. Cohen
    • 1
  • D. Hachenberger
    • 2
  1. 1.Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland (e-mail:
  2. 2.Institut für Mathematik, Universität Augsburg, D-86159 Augsburg, Germany (e-mail:

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