Abstract.
Let E be a finite degree extension over a finite field F = GF(q), G the Galois group of E over F and let a∈F 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 {w g∣g∈G} 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.
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Received: June 15, 1998; revised version: December 2, 1998
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Cohen, S., Hachenberger, D. Primitive Normal Bases with Prescribed Trace. AAECC 9, 383–403 (1999). https://doi.org/10.1007/s002000050112
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DOI: https://doi.org/10.1007/s002000050112