Kummer Spaces in Cyclic Algebras of Prime Degree

TL;DR

This paper classifies monomial Kummer subspaces in cyclic division algebras of prime degree, establishing that all such spaces are standard with a maximum dimension of p+1, impacting the understanding of their structure.

Contribution

It proves that all monomial Kummer subspaces in prime degree cyclic division algebras are standard and bounds their dimension by p+1.

Findings

All monomial Kummer subspaces are standard.

Maximum dimension of Kummer subspaces is p+1.

In generic cyclic algebras, the dimension bound applies.

Abstract

We classify the monomial Kummer subspaces of division cyclic algebras of prime degree $p$, showing that every such space is standard, and in particular the dimension is no greater than $p+1$. It follows that in a generic cyclic algebra, the dimension of any Kummer subspace is at most $p+1$.