The celebrated Primitive Normal Basis Theorem states that for any n ≥ 2 and any finite field F q , there exists an element α ∈ F q n that is simultaneously primitive and normal over F q . In this paper, we prove some variations of this result, completing the proof of a conjecture proposed by Anderson and Mullen (2014). Our results also imply the existence of elements of F q n with multiplicative order (q n − 1)/2 and prescribed trace over F q .
Let F q be the finite field of characteristic p with q elements and F q n its extension of degree n. We prove that there exists a primitive element of F q n that produces a completely normal basis of F q n over F q , provided that n = p ℓ m with (m, p) = 1 and q > m.
An extension of the primitive normal basis theorem and its strong version is proved. Namely, we show that for nearly all A = a b c d ∈ GL 2 (F q ), there exists some x ∈ F q m such that both x and (−dx + b)/(cx − a) are simultaneously primitive elements of F q m and produce a normal basis of F q m over F q , granted that q and m are large enough.
Let r, n > 1 be integers and q be any prime power q such that r | q n − 1. We say that the extension F q n /Fq possesses the line property for rprimitive elements if, for every α, θ ∈ F * q n , such that F q n = Fq(θ), there exists some x ∈ Fq, such that α(θ + x) has multiplicative order (q n − 1)/r. We prove that, for sufficiently large prime powers q, F q n /Fq possesses the line property for r-primitive elements. In particular, when r = n = 2 and (necessarily) q is odd, we show that F q 2 /Fq posseses the line property for 2-primitive elements unless q ∈ {3, 5, 7, 9, 11, 13, 31, 41}. We also discuss the (weaker) translate property for extensions.
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