论文标题
在有限特征的有限场上存在理性形式的原始正常元素的存在
On existence of primitive normal elements of rational form over finite fields of even characteristic
论文作者
论文摘要
让$ q $是偶数功率，整数是$ m \ geq2 $。由$ \ mathbb {f} _q $，我们表示订单$ q $的有限字段和$ \ m athbb {f} _ {q^m} $其扩展度$ m $。在本文中，我们研究了原始的普通对$（α，\，f（α））$的存在，其中$ f（x）= \ dfrac {ax^2+bx+bx+c} {dx+e} \ in \ in \ mathbb {f} $ f = \ begin {pmatrix} a \，＆b \，＆c \\ 0 \，＆d \，＆d \，＆e en en en en en en e \ end {pmatrix} $ $ \ in m_ {2 \ times 3}（\ fm）$是2。 \，＆1 \，＆0 \\ 0 \ \，＆1 \，＆0 \ end {pmatrix} $如果$ q = 2 $，而$ m $很奇怪，然后我们提供了可能的和真实的异常对$（q，m）$的明确列表。
Let $q$ be an even prime power and $m\geq2$ an integer. By $\mathbb{F}_q$, we denote the finite field of order $q$ and by $\mathbb{F}_{q^m}$ its extension degree $m$. In this paper we investigate the existence of a primitive normal pair $(α, \, f(α))$, with $f(x)= \dfrac{ax^2+bx+c}{dx+e} \in \mathbb{F}_{q^m}(x)$, where the rank of the matrix $F= \begin{pmatrix}a \, &b\, & c\\ 0\, &d \, &e \end{pmatrix}$ $\in M_{2 \times 3}(\Fm) $ is 2. Namely, we establish sufficient conditions to show that nearly all fields of even characteristic possess such elements, except for $\begin{pmatrix} 1 \, &1 \, & 0\\ 0\, &1 \, &0 \end{pmatrix}$ if $q=2$ and $m$ is odd, and then we provide an explicit list of possible and genuine exceptional pairs $(q,m)$.
