Generalizations of the Normal Basis Theorem of Finite Fields

A combinatorial characterization of sets of integers $\{ r_0 ,r_1 , \cdots ,r_{n - 1} \} $, with $0\leqq r_i \leqq q^n - 2$, such that $\alpha ^{r_0 } ,\alpha ^{r_1 } , \cdots ,\alpha ^{r_{n - 1} } $ form a basis of $GF( q^n )$ over $GF ( q )$ for some $\alpha \in GF( {q^n } )$ is presented. This characterization is used to prove the following generalization of the normal basis theorem for finite fields of characteristic two: Let $\lambda_0 ,\lambda_1 , \cdots ,\lambda_{n - 1} $ be integers in the range $0\leqq \lambda_i < q$, with at most one $\lambda_i $ equal to zero.Then, there exists an element $\alpha \in GF( {q^n } )$ such that $\alpha ^{\lambda_0 } ,\alpha ^{\lambda_1 q} ,\alpha ^{\lambda_2 q^2 } , \cdots ,\alpha^{\lambda_{n - 1} q^{n - 1} } $ form a bais of $GF( q^n )$ over $GF( q )$. This result, which includes the normal basis theorem as a particular case when $\lambda_0 = \lambda_1 = \cdots = \lambda_{n - 1} = 1$, is proved for all choices of $\lambda_0 ,\lambda_1 , \cdots ,\lambda_{n - 1} $ s...