An extension of the (strong) primitive normal basis theorem

An extension of the primitive normal basis theorem and its strong version is proved. Namely, we show that for nearly all $$A = {\small \left( \begin{array}{cc} a&{}b \\ c&{}d \end{array} \right) } \in \mathrm{GL}_2(\mathbb {F}_{q})$$A=abcd∈GL2(Fq), there exists some $$x\in \mathbb {F}_{q^m}$$x∈Fqm such that both $$x$$x and $$(-dx+b)/(cx-a)$$(-dx+b)/(cx-a) are simultaneously primitive elements of $$\mathbb {F}_{q^m}$$Fqm and produce a normal basis of $$\mathbb {F}_{q^m}$$Fqm over $$\mathbb {F}_q$$Fq, granted that $$q$$q and $$m$$m are large enough.

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