{
  "id": "1708.09822",
  "version": "v1",
  "published": "2017-08-31T17:13:00.000Z",
  "updated": "2017-08-31T17:13:00.000Z",
  "title": "The Structure of Hopf Algebras Acting on Galois Extensions with Dihedral Groups",
  "authors": [
    "Alan Koch",
    "Timothy Kohl",
    "Paul J. Truman",
    "Robert Underwood"
  ],
  "comment": "22 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $L/K$ be a finite separable extension of fields. The Hopf Galois structures on $L/K$ have been classified by C. Greither and B. Pareigis. In the case that $L/K$ is finite separable and Galois with non-abelian group $G$, $L/K$ admits both a classical Hopf Galois structure via the $K$-Hopf algebra $KG$ and a canonical non-classical structure via the $K$-Hopf algebra $H$. In this paper we investigate the structure of $H$ as a $K$-Hopf algebra and as a $K$-algebra. For $G$ non-abelian, we show that $KG\\not \\cong H$ as $K$-Hopf algebras, and we prove that $KG$ and $H$ have the same number of components in their Wedderburn-Artin decompositions. We specialize to the case where $K=\\mathbb Q$, and $L/\\mathbb Q$ is Galois with group $D_n$, the dihedral group of order $2n$, $n\\ge 3$. For $n=3,4$, we give necessary and sufficient conditions on $L/\\mathbb Q$ under which $\\mathbb Q D_n\\cong H$ as $K$-algebras. We include some results in the case $n$ is prime, with particular attention to $n=5$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-31T17:13:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R32",
      "16T05",
      "20B35"
    ],
    "keywords": [
      "dihedral group",
      "hopf algebras acting",
      "galois extensions",
      "classical hopf galois structure",
      "finite separable extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}