### Normal bases and primitive elements over finite fields

Giorgos Kapetanakis
2014 Finite Fields and Their Applications
Let q be a prime power, m ≥ 2 an integer and A = a b c d ∈ GL 2 (F q ), where A = ( 1 1 0 1 ) if q = 2 and m is odd. We prove an extension of the primitive normal basis theorem and its strong version. Namely, we show that, except for an explicit small list of genuine exceptions, for every q, m and A, there exists some primitive x ∈ F q m such that both x and (ax+b)/(cx+d) produce a normal basis of F q m over F q . 1. q = 2, m = 3 and A = ( 0 1 1 0 ) or A = ( 1 0 1 1 ), 2. q = 3, m = 4 and A is
more » ... nti-diagonal or 3. (q, m) is (2, 4), (4, 3) or (5, 4) and d = 0. Remark. It is interesting to notice that, not only we have no new exceptions than those appearing in Theorem 1.2, but we have no exceptions at all if all of the entries of A are non-zero. This is somehow surprising, if we consider the vast number of different tranformations that the various A's define. Also, note that the (infinite) family A = ( 1 1 0 1 ), q = 2 and m odd consists solely of genuine