Optimal embeddings for maximal orders of central simple algebras of degree 3 over number fields

TL;DR

This paper characterizes when orders of degree 3 extensions can be optimally embedded into maximal orders of a degree 3 central simple algebra over a number field, providing exact embedding criteria.

Contribution

It precisely determines the conditions for optimal embeddings of degree 3 orders into maximal orders of a central simple algebra, including the distribution among isomorphism classes.

Findings

Identifies when an order cannot be embedded into any maximal order.

Determines when an order can be embedded into exactly 1/3 or 2/3 of the maximal orders.

Provides explicit criteria for optimal embeddings based on algebraic and number-theoretic properties.

Abstract

Let $B$ be a central simple algebra of degree 3 over a number field $F$ and $K/F$ be a finite extension of degree 3. For an order $S$ of $K$, we determine exactly when $S$ cannot be optimally embedded into all maximal orders of $B$. Moreover, we further determine exactly when $S$ can be optimally embedded into $\frac{1}{3}$ isomorphism classes of maximal orders of $B$ and $\frac{2}{3}$ isomorphism classes of maximal orders of $B$ in the rest of cases.