Upper bound on the number of ramified primes for odd order solvable groups

TL;DR

This paper establishes an upper bound of approximately three times the logarithm of the group order on the number of ramified primes in tamely ramified Galois extensions of the rationals for odd order solvable groups.

Contribution

It provides the first explicit upper bound on ramified primes for odd order solvable groups, linking group order to ramification complexity.

Findings

Upper bound of 3*log(|G|) for ramified primes

Applicable to all odd order solvable groups

Advances understanding of Galois realizations with minimal ramification

Abstract

Let $G$ be a finite group and let $ram^{t}(G)$ denote the minimal positive integer $n$ such that $G$ can be realized as the Galois group of a tamely ramified extension of $\mathbb{Q}$ ramified only at $n$ finite primes. Let $d(G)$ denote the minimal non negative integer for which there exists a subset $X$ of $G$ with $d(G)$ elements such that the normal subgroup of $G$ generated by $X$ is all of $G$. It is known that $d(G)\leq ram^{t}(G)$. However, it is unknown whether or not every finite group $G$ can be realized as a Galois group of a tamely ramified extension of $\mathbb{Q}$ with exactly $d(G)$ ramified primes. We will show that $3\cdot log(|G|)$ is an upper bound for $ram^{t}(G)$ for all odd order solvable group $G$.