The Tamely Ramified Geometric Quantitative Minimal Ramification Problem

TL;DR

This paper proves a finite field version of a conjecture on counting Galois extensions with minimal ramification, using structure groups of racks, advancing understanding in algebraic number theory.

Contribution

It introduces a novel approach involving racks to analyze Galois extensions with minimal ramification over finite fields.

Findings

Established a finite field analogue of the Boston--Markin conjecture.

Developed new methods involving structure groups of racks.

Provided counts for Galois extensions with prescribed ramification.

Abstract

We prove a large finite field version of the Boston--Markin conjecture on counting Galois extensions of the rational function field with a given Galois group and the smallest possible number of ramified primes. Our proof involves a study of structure groups of (direct products of) racks.