论文标题
关于二次循环扩展的注释
A note on quadratic cyclotomic extensions
论文作者
论文摘要
本文提供了两个在任意磁场上的二次循环扩展中统一根源的特征。首先，我们将映射从$ \ mathbb {n} $介绍到$ \ mathbb {n} $描述这些根，与他们在现场上的订单紧密相关。其次，对于任何prime $ p $，我们确定最大自然数$ n $，以便$ζ_{p^n} $定义了一个二次循环扩展，而在field $ f $上。无论其特征如何，这种特征在不同领域都是统一的，并且适用于奇数甚至素数。
This paper provides two characterizations of the primitive roots of unity in quadratic cyclotomic extensions over arbitrary fields. Firstly, we introduce a mapping from $\mathbb{N}$ to $\mathbb{N}$ crucial for describing these roots, closely tied to their order over the field. Secondly, for any prime $p$, we determine the maximal natural number $n$ such that $ζ_{p^n}$ defines a quadratic cyclotomic extension over the field $F$. This characterization is uniform across different fields, regardless of their characteristic, and applies to both odd and even primes.
