Embeddings of fields into simple algebras: Generalizations and applications

Abstract

For two semi-simple algebras A and B over an arbitrary ground field F , we give a numerical criterion when Hom F -alg (A, B), the set of Falgebra homomorphisms between them, is non-empty. We also determine when the orbit set B × \ Hom F -alg (A, B) is finite and give an explicit formula for its cardinality. A few applications of main results are given.

Endomorphism algebras of QM abelian surfaces

“…The local-global principle (cf. [10, Theorem A.1] and [18]) asserts that this holds if and only if for any (finite) ramified place v of K for ∆, one has [L w :…”

“…Proof. This is a special case of [18,Theorem 1.2]. However, instead of referring to the general result, we prefer to give a direct proof for the reader's convenience.…”

Embeddings of fields into simple algebras over global fields

“…This paper is organized as follows. In Section 2 we collect and show some general embedding results over any field based on [17]. In Section 3, we give a more detailed study about embeddings of a field extension K into a central simple algebra A over a global field.…”

“…In the first part of this paper we answer the questions (Q1), (Q2) and (Q3). In [17], the third named author of the present paper studies the problem of embeddings of one semi-simple algebra into another one over an arbitrary ground field. In Section 2 we recall some results of embeddings obtained in [17] and provide some proofs of them for the reader's convenience.…”

“…In [17], the third named author of the present paper studies the problem of embeddings of one semi-simple algebra into another one over an arbitrary ground field. In Section 2 we recall some results of embeddings obtained in [17] and provide some proofs of them for the reader's convenience. These give rise to a numerical criterion for the question (Q1) over an arbitrary field; see Lemma 2.4 for the precise statement.…”