On normal bases of a finite field

In this paper irreducible polynomials $f(x)$ of degree $t$ are constructed over a finite field of characteristic $p>0$ with linearly independent roots, where the integer $t$ divides one of the numbers $p$, $q-1$, or $q+1$. Properties of normal bases of the field $F_{q^t}$ over $F_q$ formed by the roots $\{\omega_1,\dots,\omega_t\}$ of $f(x)$ are also studied. In particular, it is shown that the “multiplication table” of such a basis has the form $\omega_i\omega_j=\alpha_{i-j}\omega_i+\alpha_{j-1}\omega_j+\gamma$, $i\ne j$, $\alpha_k$, $\gamma\in F_q$.
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