Certain Unramified Metabelian Extensions Using Lemmermeyer Factorizations

TL;DR

This paper investigates the existence of unramified nonabelian extensions over number fields using discriminant conditions, generalizing Lemmermeyer's quadratic field result to broader class field theory contexts.

Contribution

It extends criteria for unramified extensions with nonabelian Galois groups, linking discriminant factorizations to the existence of such extensions beyond quadratic fields.

Findings

Characterization of discriminant conditions for unramified lifts

Classification of unramified nonabelian extensions over abelian fields

Generalization of Lemmermeyer's quadratic field result

Abstract

We study solutions to the Brauer embedding problem with restricted ramification. Suppose $G$ and $A$ are a abelian groups, $E$ is a central extension of $G$ by $A$, and $f:\text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow G$ a continuous homomorphism. We determine conditions on the discriminant of $f$ that are equivalent to the existence of an unramified lift $\widetilde{f}:\text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow E$ of $f$. As a consequence of this result, we use conditions on the discriminant of $K$ for $K/\mathbf{Q}$ abelian to classify and count unramified nonabelian extensions $L/K$ normal over $\mathbf{Q}$ where the (nontrivial) commutator subgroup of $\text{Gal}(L/\mathbf{Q})$ is contained in its center. This generalizes a result due to Lemmermeyer, which states that a quadratic field $\mathbf{Q}(\sqrt{d})$ has an unramified extension normal over $\mathbf{Q}$ with Galois group $H_8$ the quaternion group if and only if the discriminant factors $d=d_1 d_2 d_3$ as a product of three coprime discriminants, at most one of which is negative, satisfying the following condition on Legendre symbols: \[ \left(\frac{d_i d_j}{p_k}\right)=1 \] for $\{i,j,k\}=\{1,2,3\}$ and $p_i$ any prime dividing $d_i$.