A proof for a part of noncrossed product theorem

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 $\operatorname{char} k\nmid n$ and $p^3|n$.