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Unramified extensions of quadratic number fields with Galois group $2.A_n$

Joachim König

TL;DR

This work advances inverse Galois theory by constructing infinitely many quadratic fields that admit everywhere unramified Galois extensions with Galois group $2.A_n$, for infinite families of $n$. The authors synthesize central embedding problem methods with the stem cover $2.S_n^+$, control ramification through careful local analysis (via KLN specialization and the Specialization Inertia Theorem), and leverage Hilbert irreducibility to produce infinitely many distinct base fields. They also provide an explicit construction for the $n=6$ case, yielding $SL_2(\mathbb{F}_9)\cong 2.A_6$ as an unramified extension and demonstrating real quadratic fiber possibilities via twisting. Overall, the paper delivers infinitely many unramified realizations of $2.A_n$ for certain $n$ and outlines a robust framework combining function-field techniques with local-global embedding criteria to extend these results further.

Abstract

We realize infinitely many covering groups $2.A_n$ (where $A_n$ is the alternating group) as the Galois group of everywhere unramified Galois extensions over infinitely many quadratic number fields. After several predecessor works investigating special cases or proving conditional results in this direction, these are the first unramified realizations of infinitely many of these groups.

Unramified extensions of quadratic number fields with Galois group $2.A_n$

TL;DR

This work advances inverse Galois theory by constructing infinitely many quadratic fields that admit everywhere unramified Galois extensions with Galois group , for infinite families of . The authors synthesize central embedding problem methods with the stem cover , control ramification through careful local analysis (via KLN specialization and the Specialization Inertia Theorem), and leverage Hilbert irreducibility to produce infinitely many distinct base fields. They also provide an explicit construction for the case, yielding as an unramified extension and demonstrating real quadratic fiber possibilities via twisting. Overall, the paper delivers infinitely many unramified realizations of for certain and outlines a robust framework combining function-field techniques with local-global embedding criteria to extend these results further.

Abstract

We realize infinitely many covering groups (where is the alternating group) as the Galois group of everywhere unramified Galois extensions over infinitely many quadratic number fields. After several predecessor works investigating special cases or proving conditional results in this direction, these are the first unramified realizations of infinitely many of these groups.
Paper Structure (4 sections, 8 theorems, 8 equations)

This paper contains 4 sections, 8 theorems, 8 equations.

Table of Contents

  1. Statement of main result
  2. Technical tools
  3. Proof of Theorem \ref{['thm:main']}
  4. Small cases not covered by Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $n\ge 4$ be such that either $n\equiv 3 \bmod 8$ is a prime, or $n\equiv 2\bmod 8$ and $n-1$ is a prime or a square. Then there exist infinitely many quadratic number fields which possess an unramified Galois extension with Galois group $2.A_n$, the double covering group of $A_n$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • proof
  • Remark 3.1
  • Theorem 4.1
  • ...and 8 more