Normal bases and primitive elements over finite fields

Let q be a prime power, m>=2 an integer and A=(abcd)@?GL"2(F"q), 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 [email protected]?F"q"^"m such that both x and (ax+b)/(cx+d) produce a normal basis of F"q"^"m over F"q.

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