A note on the transport of (near-)field structures

TL;DR

This paper investigates how to determine all additions that turn a scalar group into a (near-)field by analyzing automorphisms and transformations, providing a structural understanding of such modifications.

Contribution

It characterizes continuous automorphisms of real and complex fields and describes how these automorphisms can be used to transport (near-)field structures.

Findings

Computed the set of continuous automorphisms of real and complex fields.

Characterized endo-bijections defining (near-)field additions.

Provided a framework for transforming scalar groups into (near-)fields.

Abstract

This paper addresses the question: given a scalar group, can we determine all the additions that transform this scalar group into a (near-)field? A key approach to addressing this problem involves transporting (near-)field structures via multiplicative automorphisms. We compute the set of continuous multiplicative automorphisms of the real and complex fields and analyze their structures. Additionally, we characterize the endo-bijections on the scalar group that define these additions.