The primitive normal basis theorem asks whether every finite field extension has a primitive normal basis of this extension. The proof of this problem has recently been completed by Lenstra and Schoof (1987) , and another proof is given by Cohen and Huczynska (2003) . We present a more general result, where the primitive element generating a normal basis is replaced by a primitive element generating the finite Carlitz module. Such generators always exist except for finitely many cases which might not exist.
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