Normal bases and primitive elements over finite fields

Let q be a prime power, m ≥ 2 an integer and A = ( a b c d ) ∈ GL2(Fq), where A 6= ( 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 ∈ Fqm such that both x and (ax+b)/(cx+d) produce a normal basis of Fqm over Fq.