Cyclic Division Algebras of Odd Prime Degree are never Amitsur-Small

TL;DR

This paper proves that cyclic division algebras of odd prime degree over their center do not satisfy the Amitsur-Small property, highlighting a fundamental limitation in their ideal structure.

Contribution

It establishes a new non-existence result for the Amitsur-Small property in a broad class of division algebras, specifically cyclic division algebras of odd prime degree.

Findings

Cyclic division algebras of odd prime degree are not Amitsur-Small.

The result applies to all such algebras over their center.

Provides insight into the ideal structure of these algebras.

Abstract

A division ring $D$ is Amitsur-Small if for every $n$ and every maximal left ideal $I$ in $D[x_1,\dots,x_n]$, $I \cap D[x_1,\dots,x_{n-1}]$ is maximal in $D[x_1,\dots,x_{n-1}]$. The goal of this note is to prove that cyclic division algebras of odd prime degree over their center are never Amitsur-Small.