Abstract
Let q be a prime power, m⩾2 an integer and A=(abcd)∈GL2(Fq), where A≠(1101) 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.
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