Crossed product algebra

Summary

A crossed product algebra is a central simple algebra built from a finite Galois extension $L/K$. The extra datum is a $2$-cocycle of its Galois group with values in $L^\times$. It contains $L$ as a maximal subfield, and each Galois automorphism is realized as conjugation by an invertible element. Every central simple algebra is Brauer equivalent to a crossed product, which gives the isomorphism between the Brauer group and a second Galois cohomology group. This algebraic notion is distinct from the crossed product of operator algebras, although both twist a ring by a group action.