A proof for a part of noncrossed product theorem

TL;DR

This paper provides a concise proof for a specific case in the noncrossed product theorem, focusing on division algebras with certain degree and characteristic conditions, advancing understanding in algebraic structures.

Contribution

It offers a short proof for a key step in noncrossed product division algebra theory under particular divisibility and characteristic assumptions.

Findings

Proof applies when characteristic does not divide n

Addresses case where p^3 divides n

Supports noncrossed product algebra construction

Abstract

The first examples of noncrossed product division algebras were given by Amitsur in 1972. His method is based on two basic steps: (1) If the universal division algebra $U(k,n)$ is a $G$-crossed product then every division algebra of degree $n$ over $k$ should be a $G$-crossed product; (2) There are two division algebras over $k$ whose maximal subfields do not have a common Galois group. In this note, we give a short proof for the second step in the case where $\chr k\nmid n$ and $p^3|n$.