标题： 代数数域中的扭曲导子 显示英文标题

标题： Twisted Derivations in Algebraic Number Fields

摘要： 设 $A$是一个有单位元的交换环，且 $B = A[\theta]$是 $A$的整扩张。 假设$B$是一个整环，其分式域为$\mathbb{K}$，而$\mathbb{E}$是$\theta$在$\mathbb{K}$上的最小分裂域。 假设$\sigma, \tau: B \rightarrow \mathbb{E}$是两个不同的环同态，它们逐元素固定$A$。 在本文中，我们对所有$A$-线性映射$D: B \rightarrow \mathbb{E}$进行分类，这些映射是$(\sigma, \tau)$-导子。 由此，我们对某些域扩张、代数数域及其代数整数环中的所有$(\sigma, \tau)$-导子进行分类。 For the ring of algebraic integers, $O_{\mathbb{K}} = \mathbb{Z}[\zeta]$ of the cyclotomic number field $\mathbb{K} = \mathbb{Q}(\zeta)$ ($\zeta$ an $n^{\text{th}}$ primitive root of unity), and a pair $(\sigma, \tau)$ of two different $\mathbb{Z}$-algebra endomorphisms of $O_{\mathbb{K}}$, we conjecture a necessary and sufficient condition for a $(\sigma, \tau)$-derivation $D:O_{\mathbb{K}} \rightarrow O_{\mathbb{K}}$ to be inner. 这是针对$n$的两种不同形式进行的：(i)$n = 2^{r}p$($r \in \mathbb{N}$和$p$为奇数有理素数)，以及 (ii)$n=p^{k}$($k \in \mathbb{N}$和$p$为任意有理素数)。 我们还猜想上述两种形式的$n$的非零外导数的存在与不存在，从而解决了$O_{\mathbb{K}}$中的扭曲导数问题。 显示英文摘要

摘要： Let $A$ be a commutative ring with unity and $B = A[\theta]$ be an integral extension of $A$. Assume that $B$ is an integral domain with quotient field $\mathbb{K}$ and $\mathbb{E}$ is the minimal splitting field of $\theta$ over $\mathbb{K}$. Suppose $\sigma, \tau: B \rightarrow \mathbb{E}$ are two different ring homomorphisms that fix $A$ element-wise. In this article, we classify all $A$-linear maps $D: B \rightarrow \mathbb{E}$ which are $(\sigma, \tau)$-derivations. Consequently, we classify all $(\sigma, \tau)$-derivations in certain field extensions, algebraic number fields, and their ring of algebraic integers. For the ring of algebraic integers, $O_{\mathbb{K}} = \mathbb{Z}[\zeta]$ of the cyclotomic number field $\mathbb{K} = \mathbb{Q}(\zeta)$ ($\zeta$ an $n^{\text{th}}$ primitive root of unity), and a pair $(\sigma, \tau)$ of two different $\mathbb{Z}$-algebra endomorphisms of $O_{\mathbb{K}}$, we conjecture a necessary and sufficient condition for a $(\sigma, \tau)$-derivation $D:O_{\mathbb{K}} \rightarrow O_{\mathbb{K}}$ to be inner. This is done for two different forms of $n$: (i) $n = 2^{r}p$ ($r \in \mathbb{N}$ and $p$ an odd rational prime), and (ii) $n=p^{k}$ ($k \in \mathbb{N}$ and $p$ any rational prime). We also conjecture the existence and non-existence of non-zero outer derivations for the above two forms of $n$, thus answering the twisted derivation problem in $O_{\mathbb{K}}$.