Primitive elements of Galois extensions of finite fields

Abstract

As is well known, ${N_q}(n) = (1/n)\sum \nolimits _{d|n} {\mu (d){q^{n/d}}}$ coincides with the number of monic irreducible polynomials of $\operatorname {GF}(q)[X]$ of degree $n$. In this note we discuss the curve $_n{{\text {N}}_X}(n)$ and the solutions of equations $_n{{\text {N}}_X}(n) = b(b \geq n)$. As a corollary of these results, we present a necessary and sufficient arithmetical condition for $R/K$ to have a primitive element.