Noncrossed products in Witt's Theorem

TL;DR

This paper classifies noncrossed product division algebras over fields like k((t)), using Witt's theorem to analyze the Brauer group, revealing that noncrossed products are prevalent for large indices.

Contribution

It provides a classification of Brauer classes containing noncrossed products over k((t)) using Witt's theorem, linking their existence to index and roots of unity.

Findings

Noncrossed products occur with density 1 for large indices.

The Brauer group splits into crossed and noncrossed parts based on character analysis.

Classification depends on the relation between index and roots of unity.

Abstract

Since Amitsur's discovery of noncrossed product division algebras more than 35 years ago, their existence over more familiar fields has been an object of investigation. Brussel's work was a culmination of this effort, exhibiting noncrossed products over the rational function field k(t) and the Laurent series field k((t)) over any global field k -- the smallest possible centers of noncrossed products. Witt's theorem gives a transparent description of the Brauer group of k((t)) as the direct sum of the Brauer group of k and the character group of the absolute Galois group of k. We classify the Brauer classes over k((t)) containing noncrossed products by analyzing the fiber over chi for each character chi in Witt's theorem. In this way, a picture of the partition of the Brauer group into crossed products/noncrossed products is obtained, which is in principle ruled solely by a relation between index and number of roots of unity. For large indices the noncrossed products occur with a "natural density" equal to 1.