The Brauer group is a fundamental concept in algebraic geometry and algebra, particularly in the study of central simple algebras. It encodes information about dividing algebras and Galois cohomology. In more precise terms, the Brauer group of a field \( K \), denoted \( \text{Br}(K) \), is defined as the group of equivalence classes of central simple algebras over \( K \) under the operation of tensor product.
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The Brauer group of a field consists of Morita equivalence classes of central simple -algebras. Multiplication is induced by tensor product, the identity is , and the inverse of is the class of the opposite algebra .