# Noncyclic Division Algebras over Fields of Brauer Dimension One

## TL;DR

This paper investigates division algebras over certain fields, showing that noncyclic algebras exist for specific degrees when the residue characteristic is 2, but all are cyclic otherwise.

## Contribution

It proves the existence of noncyclic division algebras of certain degrees over fields with residue characteristic 2, and confirms cyclicity in other cases.

## Key findings

- Noncyclic division algebras exist for degrees divisible by four when p=2.
- All division algebras are cyclic when the residue characteristic is not 2.
- Period equals index in the Brauer group of the field.

## Abstract

Let $K$ be a complete discretely valued field of rank one, with residue field $\Q_p$. It is well known that period equals index in $\Br(K)$. We prove that when $p=2$ there exist noncyclic $K$-division algebras of every $2$-power degree divisible by four. Otherwise, every $K$-division algebra is cyclic.