Mal'cev-Neumann rings and noncrossed product division algebras

TL;DR

The paper establishes conditions under which Mal'cev-Neumann rings form noncrossed product division algebras and proves their existence for specific degrees, advancing understanding of algebraic structures.

Contribution

It provides a sufficient condition for Mal'cev-Neumann rings to be noncrossed product division algebras and offers an elementary proof of their existence for certain degrees.

Findings

Identified a sufficient condition for Mal'cev-Neumann rings to be noncrossed product division algebras.

Proved the existence of noncrossed product division algebras of degree 8 and p^2 for odd primes p.

Simplified the proof of existence for these algebras using elementary methods.

Abstract

We first give a sufficient condition for a Mal'cev-Neumann ring of formal series to be a noncrossed product division algebra. This result is then used to give an elementary proof of the existence of noncrossed product division algebras (of degree 8 or degree p^2 for p any odd prime).