An element $\alpha \in {\mathbb{F}}_{q^n}$ is normal over ${\mathbb{F}}_q$ if $\mathcal{B}=\{\alpha ,{\alpha }^q,{\alpha }^{q^2},\cdots ,{\alpha }^{q^{n-1}}\}$ forms a basis of ${\mathbb{F}}_{q^n}$ as a vector space over ${\mathbb{F}}_q$. It is well known that $\alpha \in {\mathbb{F}}_{q^n}$ is normal over ${\mathbb{F}}_q$ if and only if $g_{\alpha }(x)=\alpha x^{n-1}+{\alpha }^q{x}^{n-2}+\cdots +{\alpha }^{q^{n-2}}x+{\alpha }^{q^{n-1}}$ and $x^n-1$ are relatively prime over ${\mathbb{F}}_{q^n}$, that is, the degree of their greatest common divisor in ${\mathbb{F}}_{q^n}[x]$ is 0. Using this equivalence, the notion of k-normal elements was introduced in Huczynska et al. (2013): an element $\alpha \in {\mathbb{F}}_{q^n}$ is k-normal over ${\mathbb{F}}_q$ if the greatest common divisor of the polynomials $g_{\alpha }[x]$ and $x^n-1$ in ${\mathbb{F}}_{q^n}[x]$ 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 ${\mathbb{F}}_{q^n}$ 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 ${\mathbb{F}}_{q^n}$.

一个元素 $\alpha \in {\mathbb{F}}_{q^{ñ}}$ 是正常的 ${\mathbb{F}}_q$ 如果 $\mathcal{乙}=\{\alpha ，{\alpha }^q，{\alpha }^{q^{\mathrm{2个}}}，\cdots ，{\alpha }^{q^{ñ-\mathrm{1个}}}\}$ 构成 ${\mathbb{F}}_{q^{ñ}}$ 作为向量空间 ${\mathbb{F}}_q$。众所周知$\alpha \in {\mathbb{F}}_{q^{ñ}}$ 是正常的 ${\mathbb{F}}_q$ 当且仅当 $G_{\alpha }（X）=\alpha X^{ñ-\mathrm{1个}}+{\alpha }^q{X}^{ñ-\mathrm{2个}}+\cdots +{\alpha }^{q^{ñ-\mathrm{2个}}}X+{\alpha }^{q^{ñ-\mathrm{1个}}}$ 和 $X^{ñ}-\mathrm{1个}$ 相对来说比较重要 ${\mathbb{F}}_{q^{ñ}}$，即他们最大公约数的程度 ${\mathbb{F}}_{q^{ñ}}[X]$使用等价，在Huczynska等人中引入了k正态元素的概念。（2013）：元素$\alpha \in {\mathbb{F}}_{q^{ñ}}$是k -normal over${\mathbb{F}}_q$ 如果多项式的最大公约数 $G_{\alpha }[X]$ 和 $X^{ñ}-\mathrm{1个}$ 在 ${\mathbb{F}}_{q^{ñ}}[X]$具有度k ; 因此通常意义上正常的元素为0-正常。

Huczynska等。提出了关于两人的问题$（ñ，ķ）$为此，在其中存在原始k-法线元素${\mathbb{F}}_{q^{ñ}}$ 超过 ${\mathbb{F}}_q$ 他们得到了部分结果 $ķ=\mathrm{1个}$，后来Reis和Thomson（2018）完成了本案。原始正态基础定理解决了这种情况$ķ=0$。在本文中，我们完全解决了这种情况$ķ=\mathrm{2个}$使用高斯和的估计值和计算机的使用，我们还获得了存在k-法线元素的新条件${\mathbb{F}}_{q^{ñ}}$。