Maximal orders optimal embedding of central simple algebras over number fields

TL;DR

The paper addresses embedding a degree $p$ extension $K/F$ into a central simple algebra $B$ and the refined problem of optimal embeddings into maximal orders. It develops a class-field theoretic framework using the open subgroup $GN(O)$ and the associated class field $H_{GN(O)}$, proving that the genus $\mathrm{Gen}O$ is optimally selective for an $R$-order $S\subset K$ exactly when $K\subseteq H_{GN(O)}$. A global–local selectivity sandwich is established, yielding a quantitative dichotomy: if $K\subseteq H_{GN(O)}$, exactly $1/p$ of the order types admit optimal embeddings; if not, embeddings occur for all types. These results extend Voight’s quaternionic selectivity to central simple algebras of degree $p$, clarifying how class-field data govern integral optimal embeddings into maximal orders across higher degrees.

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.

Theorems & Definitions (29)

• Theorem 1.1: Albert–Brauer–Hasse–Noether
• Theorem 1.2
• Theorem 1.3: Theorem \ref{['main']}
• Definition 2.1
• Theorem 2.1: Linowitz2012
• proof
• Theorem 2.2
• proof
• Corollary 2.3
• Lemma 2.4: voight2021quaternion