Cyclic algebra

Summary

A cyclic algebra is a crossed product algebra for a cyclic extension of fields. It is described by a generator $\sigma$ of the Galois group and one nonzero element $a$ of the base field. Every cyclic algebra of degree $2$ is a quaternion algebra, and over local and global fields every central simple algebra is cyclic. Whether a cyclic algebra splits is decided by the field norm: it splits exactly when $a$ is a norm.