Embeddings of fields into simple algebras over global fields

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 into A over F exists implies an embedding exists locally everywhere. In this paper we give detailed discussions about when the converse (i.e. the local-global principle in question) may hold.Date: November 6, 2018.

Endomorphism algebras of QM abelian surfaces

“…This is a special case of [ F is a global field, the local-global principle enters and plays a role in the problem of embeddings of simple algebras. For a detailed discussion, the reader is referred to the paper [12].…”

Characteristic Polynomials of Central Simple Algebras

“…When F is a global field and the degree [E : F ] = deg(A) is maximal, one can use the local-global principle to check the condition in Lemma 6. However, when the degree [E : F ] is not maximal, the local-global principle for embedding E in A over F fails in general; see constructions of counterexamples in [7,Sections 4 and 5]. The following lemma provides an alternative method to check this condition.…”

Embeddings of fields into simple algebras: Generalizations and applications

“…That is, if δ is not of this form, then there exist a central simple algebra A of degree δ and a finite separable field extension K with [K : F ] | δ such that the corresponding Hasse principle fails. We refer the reader to [9] for details.…”