论文标题
与理想类相对应的PID的相似性类别的代表
Representatives of similarity classes of matrices over PIDs corresponding to ideal classes
论文作者
论文摘要
对于主要的理想域$ a $，latimer--macduffee通信设置了$ \ operatatorName {m} _ {n}（a）$的相似性类别的矩阵类别，其特征性的特征性polynomial $ f（x）$与订单的理想类别$ a [x]/（x]/（f（x）））。我们证明，当$ a [x]/（f（x））$是最大的（即，封闭式封闭，即Dedekind域）时，每个相似性类都包含一个代表，从某种意义上来说，它接近伴侣矩阵。证明的第一步是证明与理想（不一定是质量）相对应的任何相似性类都包含所需形式的代表。第二步是由于Lenstra而引起的先前未发表的结果，这意味着当$ a [x]/（f（x））$最大时，每个理想类都包含一个学位的理想。
For a principal ideal domain $A$, the Latimer--MacDuffee correspondence sets up a bijection between the similarity classes of matrices in $\operatorname{M}_{n}(A)$ with irreducible characteristic polynomial $f(x)$ and the ideal classes of the order $A[x]/(f(x))$. We prove that when $A[x]/(f(x))$ is maximal (i.e., integrally closed, i.e., a Dedekind domain), then every similarity class contains a representative that is, in a sense, close to being a companion matrix. The first step in the proof is to show that any similarity class corresponding to an ideal (not necessarily prime) of degree one contains a representative of the desired form. The second step is a previously unpublished result due to Lenstra that implies that when $A[x]/(f(x))$ is maximal, every ideal class contains an ideal of degree one.
