{
  "id": "2109.02130",
  "version": "v1",
  "published": "2021-09-05T17:55:10.000Z",
  "updated": "2021-09-05T17:55:10.000Z",
  "title": "Conjugacy of Integral Matrices over Algebraic Extensions",
  "authors": [
    "Rebecca Afandi"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-05T17:55:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "integral matrices",
      "algebraic extensions",
      "algebraic integral extension",
      "principal ideal theorems",
      "class field theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}