Multiplicative quadratic maps

TL;DR

This paper characterizes multiplicative quadratic maps between rings and fields, showing they are induced by homomorphisms or composition algebras, and establishes a multiplicative version of Artin's Theorem for uniqueness.

Contribution

It provides a complete classification of multiplicative quadratic maps, revealing their structure and uniqueness properties in the context of rings and fields.

Findings

Multiplicative quadratic maps are induced by homomorphisms or composition algebras.

When K is a field, such maps are products of two field homomorphisms.

A multiplicative version of Artin's Theorem proves the uniqueness of these maps up to multiplicity.

Abstract

In this paper we prove that a multiplicative quadratic map between a unital ring $K$ and a field $L$ is induced by a homomorphism from $K$ into $L$ or a composition algebra over $L$. Especially we show that if $K$ is a field, then every multiplicative quadratic map is the product of two field homomorphisms. Moreover, we prove a multiplicative version of Artin's Theorem showing that a product of field homomorphisms is unique up to multiplicity.