Tensor product of algebras

In this chapter we introduce tensor product of K-algebras. An important property is that tensor product of central simple algebras is again a central simple algebra. We establish several algebra isomorphisms identifying tensor products of matrix algebras and linked quaternion algebras. As an application we introduce the Hasse algebra of a nonsingular symmetric bilinear space (quadratic form) and show that it is an isometry (equivalence) invariant.