Maximal orders optimal embedding of central simple algebras over number fields

TL;DR

This paper investigates the conditions under which a ring of integers from a field extension can be optimally embedded into maximal orders of a central simple algebra over a number field.

Contribution

It proves that such optimal embeddings exist for all maximal orders unless a specific selectivity condition is met, focusing on algebras of prime degree.

Findings

Optimal embeddings are possible into all maximal orders unless the selectivity condition holds.

The work extends classical embedding problems to central simple algebras of degree p.

Provides criteria for when optimal embeddings are obstructed by selectivity.

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.