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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 isdoi:10.1016/j.ffa.2013.12.002 fatcat:fpkf5jcvmracneyrp53ziu6xam