Normal bases and primitive elements over finite fields

Let q   be a prime power, m⩾2m⩾2 an integer and A=(abcd)∈GL2(Fq), where A≠(1101) if q=2q=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∈Fqmx∈Fqm such that both x   and (ax+b)/(cx+d)(ax+b)/(cx+d) produce a normal basis of FqmFqm over FqFq.