For any finite Galois field extension $\mathsf{K}/\mathsf{F}$, with Galois group $G = \mathrm{Gal}(\mathsf{K}/\mathsf{F})$, there exists an element $\alpha \in \mathsf{K}$ whose orbit $G\cdot\alpha$ forms an $\mathsf{F}$-basis of $\mathsf{K}$. Such an $\alpha$ is called a normal element and $G\cdot\alpha$ is a normal basis. We introduce a probabilistic algorithm for testing whether a given $\alpha \in \mathsf{K}$ is normal, when $G$ is either a finite abelian or a metacyclic group. The algorithm is based on the fact that deciding whether $\alpha$ is normal can be reduced to deciding whether $\sum_{g \in G} g(\alpha)g \in \mathsf{K}[G]$ is invertible; it requires a slightly subquadratic number of operations. Once we know that $\alpha$ is normal, we show how to perform conversions between the working basis of $\mathsf{K}/\mathsf{F}$ and the normal basis with the same asymptotic cost.

中文翻译：

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正规基的次二次时间算法

对于任何有限伽罗瓦域扩展 $\mathsf{K}/\mathsf{F}$，有伽罗瓦群 $G = \mathrm{Gal}(\mathsf{K}/\mathsf{F})$，存在一个元素$\alpha \in \mathsf{K}$ 的轨道 $G\cdot\alpha$ 形成了 $\mathsf{K}$ 的 $\mathsf{F}$ 基。这样的 $\alpha$ 称为正规元素，$G\cdot\alpha$ 是正规基。我们引入了一种概率算法来测试给定的 $\alpha \in \mathsf{K}$ 是否正常，当 $G$ 是有限阿贝尔群或元循环群时。该算法基于判断$\alpha$是否正常可以简化为判断$\sum_{g \in G} g(\alpha)g \in \mathsf{K}[G]$是否可逆; 它需要稍微次二次的操作数。一旦我们知道 $\alpha$ 是正常的，