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

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.