A Direct Approach to Noncrossed Product Division Algebras

TL;DR

This paper introduces a valuation theoretic method to construct explicit noncrossed product division algebras over small fields, providing concrete examples and structure constants, advancing the understanding of these algebraic structures.

Contribution

It presents a new valuation theoretic approach to directly construct noncrossed product division algebras over small fields, with explicit examples and structure constants.

Findings

Constructed explicit noncrossed product division algebras

Provided examples as iterated twisted function fields

Enabled explicit computation of structure constants

Abstract

A valuation theoretic approach is presented that directly leads to division algebras that are noncrossed products (instead of, e.g., describing Brauer classes of noncrossed products in an abstract manner). While this feature is shared by Amitsur's original construction, the new approach works over small fields. It is further demonstrated how it can be used to obtain very explicit examples of noncrossed products in the form of iterated twisted function fields over division algebras over global fields. The examples allow even to write down structure constants of noncrossed products.