代数扩张上整矩阵的共轭性
Conjugacy of Integral Matrices over Algebraic Extensions
摘要：
本文研究了具有子环$\mathbb{Z}$的环$R$的整矩阵与$\text{GL}_{n}(R)$中元素的共轭性.我们注意到，哈塞原理在矩阵共轭的上下文中不成立，因为对于所有$p$都是$\text{GL}_{n}(\mathbb{Z}_p)$ -共轭的矩阵不一定是$\text{GL}_{n}(\mathbb{Z})$ -共轭的。通过Guralnick定理，我们知道整数$n \times n$矩阵对所有素数$p$是$\text{GL}_{n}(\mathbb{Z}_p)$ -共轭的当且仅当它们被$\mathbb{Z}$的某个代数整数扩张$E$的$\text{GL}_{n}(E)$中的元素共轭.我们研究的问题，找到这个扩展$E$。由于Latimer和MacDuffee关于描述$\mathbb{Z}$ -共轭性的一个结果可以推广到$R$ -共轭性的上下文中，对于$R$的任何整环，我们可以将现有的$\mathbb{Z}$ -共轭性的算法应用到新的上下文中。我们还提供了一种方法来寻找$E$，它利用类域理论的主要理想定理。我们用几个例子来说明我们的方法。
英文摘要：
We consider conjugacy of integral matrices by elements in $\text{GL}_{n}(R)$ for certain rings $R$ with subring $\mathbb{Z}$. We note that a Hasse principal does not hold in the context of matrix conjugacy because matrices which are $\text{GL}_{n}(\mathbb{Z}_p)$-conjugate for all $p$ are not necessarily $\text{GL}_{n}(\mathbb{Z})$-conjugate. By a theorem of Guralnick, we know that integral $n \times n$ matrices are $\text{GL}_{n}(\mathbb{Z}_p)$-conjugate for all primes $p$ if and only if they are conjugate by an element in $\text{GL}_{n}(E)$ for some algebraic integral extension $E$ of $\mathbb{Z}$. We study the problem of finding this extension $E$. Since a result by Latimer and MacDuffee for describing $\mathbb{Z}$-conjugacy can be generalized to the context of $R$-conjugacy for $R$ any integral domain, we can adapt an existing algorithm for $\mathbb{Z}$-conjugacy to a new context. We also offer a method for finding $E$ which makes use of the principal ideal theorems of class field theory. We illustrate our method in several examples.
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