---
_id: '65358'
abstract:
- lang: eng
  text: We determine the asymptotic growth of extensions of local function fields
    of characteristic p counted by discriminant, where the Galois group is a subgroup
    of the affine group AGL_1(p). More general, we solve the corresponding counting
    problems for all groups which arise in a tower of a cyclic extension of order
    p over a cyclic extension of degree d coprime to p. This in particular give answers
    for certain non-abelian groups including S_3, dihedral groups of order 2p, and
    many Frobenius groups.
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
- first_name: Raphael
  full_name: Müller, Raphael
  id: '55246'
  last_name: Müller
citation:
  ama: Klüners J, Müller R. Counting Frobenius extensions over local function fields.
    <i>arXiv:260402152</i>. Published online 2026.
  apa: Klüners, J., &#38; Müller, R. (2026). Counting Frobenius extensions over local
    function fields. In <i>arXiv:2604.02152</i>.
  bibtex: '@article{Klüners_Müller_2026, title={Counting Frobenius extensions over
    local function fields}, journal={arXiv:2604.02152}, author={Klüners, Jürgen and
    Müller, Raphael}, year={2026} }'
  chicago: Klüners, Jürgen, and Raphael Müller. “Counting Frobenius Extensions over
    Local Function Fields.” <i>ArXiv:2604.02152</i>, 2026.
  ieee: J. Klüners and R. Müller, “Counting Frobenius extensions over local function
    fields,” <i>arXiv:2604.02152</i>. 2026.
  mla: Klüners, Jürgen, and Raphael Müller. “Counting Frobenius Extensions over Local
    Function Fields.” <i>ArXiv:2604.02152</i>, 2026.
  short: J. Klüners, R. Müller, ArXiv:2604.02152 (2026).
date_created: 2026-04-07T08:13:59Z
date_updated: 2026-04-07T08:14:45Z
external_id:
  arxiv:
  - '2604.02152'
language:
- iso: eng
publication: arXiv:2604.02152
status: public
title: Counting Frobenius extensions over local function fields
type: preprint
user_id: '82981'
year: '2026'
...
---
_id: '55192'
abstract:
- lang: eng
  text: "We describe the group of $\\mathbb Z$-linear automorphisms of the ring of\r\nintegers
    of a number field $K$ that preserve the set $V_{K,k}$ of $k$th\r\npower-free integers:
    every such map is the composition of a field automorphism\r\nand the multiplication
    by a unit.\r\n  We show that those maps together with translations generate the
    extended\r\nsymmetry group of the shift space $\\mathbb D_{K,k}$ associated to
    $V_{K,k}$.\r\nMoreover, we show that no two such dynamical systems $\\mathbb D_{K,k}$
    and\r\n$\\mathbb D_{L,l}$ are topologically conjugate and no one is a factor system
    of\r\nanother.\r\n  We generalize the concept of $k$th power-free integers to
    sieves and study\r\nthe resulting admissible shift spaces."
author:
- first_name: Fabian
  full_name: Gundlach, Fabian
  id: '100450'
  last_name: Gundlach
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Gundlach F, Klüners J. Symmetries of power-free integers in number fields and
    their shift  spaces. <i>arXiv:240708438</i>. Published online 2024.
  apa: Gundlach, F., &#38; Klüners, J. (2024). Symmetries of power-free integers in
    number fields and their shift  spaces. In <i>arXiv:2407.08438</i>.
  bibtex: '@article{Gundlach_Klüners_2024, title={Symmetries of power-free integers
    in number fields and their shift  spaces}, journal={arXiv:2407.08438}, author={Gundlach,
    Fabian and Klüners, Jürgen}, year={2024} }'
  chicago: Gundlach, Fabian, and Jürgen Klüners. “Symmetries of Power-Free Integers
    in Number Fields and Their Shift  Spaces.” <i>ArXiv:2407.08438</i>, 2024.
  ieee: F. Gundlach and J. Klüners, “Symmetries of power-free integers in number fields
    and their shift  spaces,” <i>arXiv:2407.08438</i>. 2024.
  mla: Gundlach, Fabian, and Jürgen Klüners. “Symmetries of Power-Free Integers in
    Number Fields and Their Shift  Spaces.” <i>ArXiv:2407.08438</i>, 2024.
  short: F. Gundlach, J. Klüners, ArXiv:2407.08438 (2024).
date_created: 2024-07-12T08:16:37Z
date_updated: 2024-07-12T08:19:11Z
external_id:
  arxiv:
  - '2407.08438'
language:
- iso: eng
publication: arXiv:2407.08438
status: public
title: Symmetries of power-free integers in number fields and their shift  spaces
type: preprint
user_id: '82981'
year: '2024'
...
---
_id: '57218'
abstract:
- lang: eng
  text: "We arrange the orders in an algebraic number field in a tree. This tree can\r\nbe
    used to enumerate all orders of bounded index in the maximal order as well\r\nas
    the orders over some given order."
author:
- first_name: Markus
  full_name: Kirschmer, Markus
  last_name: Kirschmer
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
citation:
  ama: Kirschmer M, Klüners J. Enumerating orders in number fields. <i>arXiv:241108568</i>.
    Published online 2024.
  apa: Kirschmer, M., &#38; Klüners, J. (2024). Enumerating orders in number fields.
    In <i>arXiv:2411.08568</i>.
  bibtex: '@article{Kirschmer_Klüners_2024, title={Enumerating orders in number fields},
    journal={arXiv:2411.08568}, author={Kirschmer, Markus and Klüners, Jürgen}, year={2024}
    }'
  chicago: Kirschmer, Markus, and Jürgen Klüners. “Enumerating Orders in Number Fields.”
    <i>ArXiv:2411.08568</i>, 2024.
  ieee: M. Kirschmer and J. Klüners, “Enumerating orders in number fields,” <i>arXiv:2411.08568</i>.
    2024.
  mla: Kirschmer, Markus, and Jürgen Klüners. “Enumerating Orders in Number Fields.”
    <i>ArXiv:2411.08568</i>, 2024.
  short: M. Kirschmer, J. Klüners, ArXiv:2411.08568 (2024).
date_created: 2024-11-19T07:49:07Z
date_updated: 2024-11-19T07:49:48Z
external_id:
  arxiv:
  - '2411.08568'
language:
- iso: eng
publication: arXiv:2411.08568
status: public
title: Enumerating orders in number fields
type: preprint
user_id: '82981'
year: '2024'
...