Let $p$ be an odd prime number, and assume that $2$ is a primitive root modulo $p^2$. Then the $p^m$th cyclotomic polynomial $Φ_{p^m}$ is irreducible over the Galois field GF$(2)$ for all $m \geq 1$. If $m \geq 1$ and $α_m$ is a root of $Φ_{p^{m+1}}$, then it is proved that $w_m:=\sum_{j=0}^m α_m^{p^i}$ generates a normal basis for $E:= GF(2^{(p-1)p^m})$ over $F:= GF(2)$ (this is the main result of the paper). The proof is based on a simple criterion for normal bases which requires the determination of the $(E,F)$-trace of the elements $w_mw_m^{2^j}$ for $j=0,\dots, (p-1)p^m-1$. The assumptions are used to establish this data. Moreover, for the case $p=3$ the trace-dual basis corresponding to $w_m$ can be determined explicitly. The explicit determination of normal bases is an interesting topic which is important for various applications involving arithmetic computations in field extensions presented with respect to a normal basis. Justifying the assumptions, this is in particular the case when the ground field is $GF(2)$, the field with two elements. However, the main result can be generalized considerably. E.g., assuming that $q>1$ is a prime power which is a primitive root modulo $p^2$, then $GF(2)$ can be replaced by $GF(q)$. The proof however uses more involved techniques by considering the module structure of the extension field with respect to the ground field and the corresponding Frobenius automorphism. For more details, we like to refer to the reviewer’s recent monograph, which constitutes an extensive treatment of (explicit constructions of) normal bases over finite fields [{\it D. Hachenberger}, Finite fields: normal bases and completely free elements, Kluwer Academic Publishers (1997; Zbl 0864.11065)].
Reviewer: D.Hachenberger (Augsburg)