# Unramified extensions over low degree number fields

## TL;DR

This paper proves the existence of certain unramified Galois extensions over low degree number fields for various nonsolvable groups, providing evidence for a broader conjecture about all finite groups.

## Contribution

It establishes the existence of unramified Galois extensions with prescribed Galois groups over low degree number fields and function fields, advancing the understanding of the inverse Galois problem.

## Key findings

- Existence of extensions with Galois group G and controlled ramification over $\,Q$
- Construction of unramified Galois extensions over quadratic fields for certain groups
- Evidence supporting the conjecture for all finite groups via function field analogues.

## Abstract

For various nonsolvable groups $G$, we prove the existence of extensions of the rationals $\mathbb{Q}$ with Galois group $G$ and inertia groups of order dividing $ge(G)$, where $ge(G)$ is the smallest exponent of a generating set for $G$. For these groups $G$, this gives the existence of number fields of degree $ge(G)$ with an unramified $G$-extension. The existence of such extensions over $\mathbb{Q}$ for all finite groups would imply that, for every finite group $G$, there exists a quadratic number field admitting an unramified $G$-extension, as was recently conjectured. We also provide further evidence for the existence of such extensions for all finite groups, by proving their existence when $\mathbb{Q}$ is replaced with a function field $k(t)$ where $k$ is an ample field.