The Golomb topology of polynomial rings, II

TL;DR

This paper explores the Golomb topology on polynomial rings over fields, revealing that the characteristic of the field is a topological invariant and characterizing self-homeomorphisms of the Golomb space.

Contribution

It establishes the invariance of field characteristic under Golomb topology and describes all self-homeomorphisms of the Golomb space for polynomial rings over infinite fields.

Findings

Characteristic of the field is a topological invariant.

Self-homeomorphisms are compositions of units and ring automorphisms.

Density of irreducible polynomials in the Golomb space.

Abstract

We study the interplay of the Golomb topology and the algebraic structure in polynomial rings $K[X]$ over a field $K$. In particular, we focus on infinite fields $K$ of positive characteristic such that the set of irreducible polynomials of $K[X]$ is dense in the Golomb space $G(K[X])$. We show that, in this case, the characteristic of $K$ is a topological invariant, and that any self-homeomorphism of $G(K[X])$ is the composition of multiplication by a unit and a ring automorphism of $K[X]$.