Maximal orders optimal embedding of central simple algebras over number fields

Abstract

Given a number field $F$ and $R$ be the ring of integers of $F$, the problem of embedding a field extension $K/F$ into a central simple algebra $B$ is classical. This paper proves that when the central simple algebra has degree $p$, the $R$-order $S\subset K$ can be optimal embedded into all maximal $R$-orders $O\subset B$, unless satisfies the optimal selectivity condition.