Cyclic Division Algebras of Odd Prime Degree are never Amitsur-Small
Adam Chapman, Ilan Levin, Marco Zaninelli
TL;DR
This work proves that cyclic division algebras of odd prime degree $p$ over their center are not Amitsur-Small. It combines structural results on cyclic algebras $D=(K/F,\beta)$, including the norm criterion for division and a $p$-special closure $L$ to ensure nonisomorphic maximal subfields $L[i]$ and $L[j]$, with a Chapman-Paran construction in $D[x,y]$ using $f$, the minimal polynomial of $i$, and the commutation with $j$ to produce a maximal left ideal $I=ig\langle f,\,y-j\big\rangle$ whose intersection with $D[x]$ is not maximal. This explicit obstruction shows that odd prime degree cyclic division algebras cannot be Amitsur-Small, completing the understanding for this family and aligning with related results for degree $3$ cases. The work highlights how subfield embeddings and norm criteria interact with noncommutative polynomial rings to determine maximal left-ideal behavior.
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.