A generalization of the primitive normal basis theorem
Let \(A=\mathbb F_q[T]\) and fix irreducible \(f\in A\) of degree \(d\). Set \(E_m=\mathbb F_{q^{dm}}\). One can define an \(A\)-module structure on \(E_m\) resulting in the Carlitz module \(C(E_m)\). \(C(E_m)\) is isomorphic, as an \(A\)-module, to \(A/(f^m-1)\) and so is cyclic. A primitive generator for \(C(E_m)\) is a primitive element \(\alpha\in E_m\) that generates \(C(E_m)\). The authors prove that \(C(E_m)\) has a primitive generator except possibly for finitely many \((q, d, m)\), which are given explicitly. They note there is no primitive generator for \((q,d,m)=(2,2,1)\). When \(f=T\), a primitive generator of \(C(E_m)\) yields a primitive normal basis for \(E_m\) over \(E_1\). Lenstra and Schoof have shown that primitive normal bases exist in all cases.
scientific article; zbMATH DE number 5817571