Minimality of topological matrix groups and Fermat primes

TL;DR

This paper investigates the minimality of certain topological matrix groups over local fields, introduces new criteria for minimality, and characterizes Fermat primes through properties of special linear groups with p-adic topologies.

Contribution

It establishes the minimality of the special upper triangular group over local fields and provides new criteria for minimality of special linear groups, linking these to Fermat primes.

Findings

$SUT(n, ext{F})$ is minimal for all local fields $ ext{F}$ of characteristic not 2.

Criteria for minimality and total minimality of $SL(n, ext{F})$ are developed.

Fermat primes are characterized via minimality of $SL(p-1, ext{Q})$ and $SL(p-1, ext{Q}(i))$.

Abstract

Our aim is to study topological minimality of some natural matrix groups. We show that the special upper triangular group $SUT(n,\mathbb{F})$ is minimal for every local field $\mathbb{F}$ of characteristic $\neq 2$. This result is new even for the field $\mathbb{R}$ of reals and it leads to some important consequences. We prove criteria for the minimality and total minimality of the special linear group $SL(n,\mathbb{F})$, where $\mathbb{F}$ is a subfield of a local field. This extends some known results of Remus-Stoyanov (1991) and Bader-Gelander (2017). One of our main applications is a characterization of Fermat primes, which asserts that for an odd prime $p$ the following conditions are equivalent: $\bullet$ $p$ is a Fermat prime; $\bullet$ $SL(p-1,\mathbb{Q})$ is minimal, where $\mathbb{Q}$ is the field of rationals equipped with the $p$-adic topology; $\bullet$ $SL(p-1,\mathbb{Q}(i))$ is minimal, where $\mathbb{Q}(i) \subset \mathbb{C}$ is the Gaussian rational field.