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Existence of primitive $2$-normal elements in finite fields (2007.11169v2)

Published 22 Jul 2020 in math.NT

Abstract: An element $\alpha \in \mathbb{F}{qn}$ is normal over $\mathbb{F}_q$ if $\mathcal{B}={\alpha, \alphaq, \alpha{q2}, \cdots, \alpha{q{n-1}}}$ forms a basis of $\mathbb{F}{qn}$ as a vector space over $\mathbb{F}q$. It is well known that $\alpha \in \mathbb{F}{qn}$ is normal over $\mathbb{F}q$ if and only if $g{\alpha}(x)=\alpha x{n-1}+\alphaq x{n-2}+ \cdots + \alpha{q{n-2}}x+\alpha{q{n-1}}$ and $xn-1$ are relatively prime over $\mathbb{F}{qn}$, that is, the degree of their greatest common divisor in $\mathbb{F}{qn}[x]$ is $0$. Using this equivalence, the notion of $k$-normal elements was introduced in Huczynska et al. ($2013$): an element $\alpha \in \mathbb{F}{qn}$ is $k$-normal over $\mathbb{F}_q$ if the greatest common divisor of the polynomials $g{\alpha}[x]$ and $xn-1$ in $\mathbb{F}{qn}[x]$ has degree $k$; so an element which is normal in the usual sense is $0$-normal. Huczynska et al. made the question about the pairs $(n,k)$ for which there exist primitive $k$-normal elements in $\mathbb{F}{qn}$ over $\mathbb{F}q$ and they got a partial result for the case $k=1$, and later Reis and Thomson ($2018$) completed this case. The Primitive Normal Basis Theorem solves the case $k=0$. In this paper, we solve completely the case $k=2$ using estimates for Gauss sum and the use of the computer, we also obtain a new condition for the existence of $k$-normal elements in $\mathbb{F}{qn}$.

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