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  In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finitedimensional associative algebra A, which is simple, and for which the center is exactly K. As an example, note that any simple algebra is a central simple algebra over its center. For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R). The quaternions H form a 4dimensional CSA over R, and in fact represent the only nontrivial element of the Brauer group of the reals (see below).
 In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finitedimensional A, which is simple, and for which the center is exactly K. As an example, note that any simple algebra is a central simple algebra over its center. For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R). The quaternions H form a 4dimensional CSA over R, and in fact represent the only nontrivial element of the Brauer group of the reals (see below).
 In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finitedimensional associative Kalgebra A, which is simple, and for which the center is exactly K. As an example, note that any simple algebra is a central simple algebra over its center. For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R). The quaternions H form a 4dimensional CSA over R, and in fact represent the only nontrivial element of the Brauer group of the reals (see below).

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  In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finitedimensional associative algebra A, which is simple, and for which the center is exactly K. As an example, note that any simple algebra is a central simple algebra over its center. For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R). The quaternions H form a 4dimensional CSA over R, and in fact represent the only nontrivial element of the Brauer group of the reals (see below). Given two central simple algebras A ~ M(n,S) and B ~ M(m,T) over the same field F, A and B are called similar (or Brauer equivalent) if their division rings S and T are isomorphic. The set of all equivalence classes of central simple algebras over a given field F, under this equivalence relation, can be equipped with a group operation given by the tensor product of algebras. The resulting group is called the Brauer group Br(F) of the field F. It is always a torsion group.
 In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finitedimensional A, which is simple, and for which the center is exactly K. As an example, note that any simple algebra is a central simple algebra over its center. For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R). The quaternions H form a 4dimensional CSA over R, and in fact represent the only nontrivial element of the Brauer group of the reals (see below). Given two central simple algebras A ~ M(n,S) and B ~ M(m,T) over the same field F, A and B are called similar (or Brauer equivalent) if their division rings S and T are isomorphic. The set of all equivalence classes of central simple algebras over a given field F, under this equivalence relation, can be equipped with a group operation given by the tensor product of algebras. The resulting group is called the Brauer group Br(F) of the field F. It is always a torsion group.
 In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finitedimensional associative Kalgebra A, which is simple, and for which the center is exactly K. As an example, note that any simple algebra is a central simple algebra over its center. For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R). The quaternions H form a 4dimensional CSA over R, and in fact represent the only nontrivial element of the Brauer group of the reals (see below). Given two central simple algebras A ~ M(n,S) and B ~ M(m,T) over the same field F, A and B are called similar (or Brauer equivalent) if their division rings S and T are isomorphic. The set of all equivalence classes of central simple algebras over a given field F, under this equivalence relation, can be equipped with a group operation given by the tensor product of algebras. The resulting group is called the Brauer group Br(F) of the field F. It is always a torsion group.

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