Embeddings of fields in simple algebras over global fields

TL;DR

This paper investigates the conditions under which local embeddings of fields into simple algebras over global fields imply a global embedding, exploring the validity of the local-global principle.

Contribution

It provides a detailed analysis of the local-global principle for embeddings of fields into simple algebras over global fields, identifying when the principle holds or fails.

Findings

Conditions for the local-global principle to hold

Counterexamples where the principle fails

Criteria for global embeddings based on local data

Abstract

Let $F$ be a global field, $A$ a central simple algebra over $F$ and $K$ a finite (separable or not) field extension of $F$ with degree $[K:F]$ dividing the degree of $A$ over $F$. An embedding of $K$ in $A$ over $F$ exists implies an embedding exists locally everywhere. In this paper we give detailed discussions when the converse (i.e. the local-global principle in question) may hold.