The Golomb topology of polynomial rings

TL;DR

This paper investigates the Golomb topology on polynomial rings over fields, establishing conditions under which the topological spaces are homeomorphic, especially focusing on fields that are algebraic extensions of finite fields.

Contribution

It provides a characterization of when Golomb spaces of polynomial rings over different fields are homeomorphic, linking topological properties to field isomorphisms.

Findings

Golomb spaces of polynomial rings over algebraic extensions of finite fields are homeomorphic iff the fields are isomorphic.

Identifies specific conditions for homeomorphism based on field characteristics.

Advances understanding of the topological classification of polynomial ring spaces.

Abstract

We study properties of the Golomb topology on polynomial rings over fields, in particular trying to determine conditions under which two such spaces are not homeomorphic. We show that if $K$ is an algebraic extension of a finite field and $K'$ is a field of the same characteristic, then the Golomb spaces of $K[X]$ and $K'[X]$ are homeomorphic if and only if $K$ and $K'$ are isomorphic.