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

L. Boonzaaier, S. Marques

TL;DR

This work studies how to recover or transport (near-)field structures from a fixed multiplicative framework by using multiplicative automorphisms. The main approach is to transport ring structures via bijections and to analyze continuous multiplicative automorphisms of key fields, especially $\mathbb{R}$ and $\mathbb{C}$, to produce new additions that turn the scalar group into a near-field. Key contributions include explicit descriptions of the continuous multiplicative automorphisms of $\mathbb{R}$ and $\mathbb{C}$, a detailed semidirect-product description of $\operatorname{Aut}_{\operatorname{cont}}(\mathbb{C},\cdot)$, and a framework for constructing additive structures on a fixed scalar group via near-field addition maps $\rho$. The results illuminate how transport of structure yields new field-like algebras and provide a toolbox for exploring near-vector spaces and related algebraic constructs, with explicit connections between additive automorphisms, exponential maps, and complex conjugation.

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.

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

TL;DR

This work studies how to recover or transport (near-)field structures from a fixed multiplicative framework by using multiplicative automorphisms. The main approach is to transport ring structures via bijections and to analyze continuous multiplicative automorphisms of key fields, especially and , to produce new additions that turn the scalar group into a near-field. Key contributions include explicit descriptions of the continuous multiplicative automorphisms of and , a detailed semidirect-product description of , and a framework for constructing additive structures on a fixed scalar group via near-field addition maps . The results illuminate how transport of structure yields new field-like algebras and provide a toolbox for exploring near-vector spaces and related algebraic constructs, with explicit connections between additive automorphisms, exponential maps, and complex conjugation.

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.
Paper Structure (10 sections, 9 theorems, 33 equations)

This paper contains 10 sections, 9 theorems, 33 equations.

Table of Contents

  1. Transport of ring structures
  2. The multiplicative automorphisms of certain fields
  3. Review of some basic results
  4. Multiplicative automorphisms of $\mathbb{C}$
  5. From $\operatorname{Aut}_{\operatorname{cont}} (\mathbb{C}, + )^{{2\pi i \mathbb{Z}} }$ to $\operatorname{Aut}_{\operatorname{cont}} ( \mathbb{C},\cdot)$
  6. Computing $\operatorname{Aut}_{\operatorname{cont}} (\mathbb{C}, + )^{{2\pi i \mathbb{Z}} }$
  7. Computing the multiplicative automorphisms of $\mathbb{C}$
  8. Identifying the group structure of the multiplicative automorphisms of $\mathbb{C}$
  9. Complex fields induced by bijective map
  10. Characterizing additive structures on a fixed scalar group

Key Result

Proposition 1.4

Let $(F, + , \cdot)$ be a near-field of characteristic $p$ and $\phi$ be an endo-bijection on $F$. The following assertions are equivalent:

Theorems & Definitions (24)

  • Definition 1.1
  • Example 1.3
  • Proposition 1.4
  • proof
  • Definition 1.5
  • Lemma 1.6
  • proof
  • Corollary 1.7
  • Lemma 2.1
  • proof
  • ...and 14 more