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.
"Embeddings of fields into simple algebras over global fields." Asian J. Math. 18 (2) 365 - 386, April 2014.