论文标题
生成$ \ mathbb {f} _ {p} $的二次扩展的非零元素
Generating the Group of Nonzero Elements of a Quadratic Extension of $\mathbb{F}_{p}$
论文作者
论文摘要
众所周知，如果$ \ mathbb {f} $是一个有限字段，则$ \ mathbb {f^{*}} $，$ \ mathbb {f} $的非零元素的集合是一个环状组。在本文中，我们将假设$ \ mathbb {f} = \ mathbb {f} _ {p} $（带有p元素的有限字段，p a prime）和$ \ mathbb {\ mathbb {\ mathbb {f}} _ {p^{2}} $ quadratic Extiense在这种情况下，组$ \ mathbb {f} _ {p}^{*} $和$ \ mathbb {f} _ {p^{2}}}^{*} $分别具有订单$ p-1 $和$ p-1 $和$ p^{2} -1 $。我们将为\ Mathbb {f} _ {p^{2}}}^{*} $的元素$ u \提供必要的条件。具体而言，我们将证明$ u $是$ \ mathbb {f} _ {p^{2}}}}^{*} $的生成器，仅当$ n（u）$ gentrates $ \ mathbb {f} _ {f} _ {p} _ {p}^{*}^{*} $ and $ \ frac { ker $ \，n $，其中$ n：\ mathbb {f} _ {p^{2}}}}^{*} \ rightArrow \ Mathbb {f} _ {p}^{*}^{*} $表示标准映射。我们还将提供一种方法来确定$ u $是否不是ker $ \，n $的生成器。
It is well known that if $\mathbb{F}$ is a finite field then $\mathbb{F^{*}}$, the set of non zero elements of $\mathbb{F}$, is a cyclic group. In this paper we will assume $\mathbb{F}=\mathbb{F}_{p}$ (the finite field with p elements, p a prime) and $\mathbb{\mathbb{F}}_{p^{2}}$ is a quadratic extension of $\mathbb{F}_{p}$. In this case, the groups $\mathbb{F}_{p}^{*}$ and $\mathbb{F}_{p^{2}}^{*}$ have orders $p-1$ and $p^{2}-1$ respectively. We will provide necessary and sufficient conditions for an element $u\in\mathbb{F}_{p^{2}}^{*}$ to be a generator. Specifically, we will prove $u$ is a generator of $\mathbb{F}_{p^{2}}^{*}$ if and only if $N(u)$ generates $\mathbb{F}_{p}^{*}$ and $\frac{u^{2}}{N(u)}$ generates Ker$\,N$, where $N:\mathbb{F}_{p^{2}}^{*}\rightarrow\mathbb{F}_{p}^{*}$ denotes the norm map. We will also provide a method for determining if $u$ is not a generator of Ker$\,N$.
