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The existence of primitive normal elements of quadratic forms over finite fields (2001.06977v1)
Published 20 Jan 2020 in math.NT
Abstract: For $q=3r$ ($r>0$), denote by $\mathbb{F}q$ the finite field of order $q$ and for a positive integer $m\geq2$, let $\mathbb{F}{qm}$ be its extension field of degree $m$. We establish a sufficient condition for existence of a primitive normal element $\alpha$ such that $f(\alpha)$ is a primitive element, where $f(x)= ax2+bx+c$, with $a,b,c\in \mathbb{F}_{qm}$ satisfying $b2\neq ac$ in $\Fm$ except for at most 9 exceptional pairs $(q,m)$.