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In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative algebra ''A'', which is simple, and for which the center is exactly ''K''. In other words, 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 4-dimensional CSA over R, and in fact represent the only non-trivial 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''.〔Lorenz (2008) p.159〕 It is always a torsion group.〔Lorenz (2008) p.194〕 ==Properties== * According to the Artin–Wedderburn theorem a finite-dimensional simple algebra ''A'' is isomorphic to the matrix algebra ''M''(''n'',''S'') for some division ring ''S''. Hence, there is a unique division algebra in each Brauer equivalence class.〔Lorenz (2008) p.160〕 * Every automorphism of a central simple algebra is an inner automorphism (follows from Skolem–Noether theorem). * The dimension of a central simple algebra as a vector space over its centre is always a square: the degree is the square root of this dimension.〔Gille & Szamuely (2006) p.21〕 The Schur index of a central simple algebra is the degree of the equivalent division algebra:〔Lorenz (2008) p.163〕 it depends only on the Brauer class of the algebra.〔Gille & Szamuely (2006) p.100〕 * The period or exponent of a central simple algebra is the order of its Brauer class as an element of the Brauer group. It is a divisor of the index,〔Jacobson (1996) p.60〕 and the two numbers are composed of the same prime factors.〔Jacobson (1996) p.61〕〔Gille & Szamuely (2006) p.104〕 * If ''S'' is a simple subalgebra of a central simple algebra ''A'' then dim''F'' ''S'' divides dim''F'' ''A''. * Every 4-dimensional central simple algebra over a field ''F'' is isomorphic to a quaternion algebra; in fact, it is either a two-by-two matrix algebra, or a division algebra. * If ''D'' is a central division algebra over ''K'' for which the index has prime factorisation :: :then ''D'' has a tensor product decomposition :: :where each component ''D''''i'' is a central division algebra of index , and the components are uniquely determined up to isomorphism.〔Gille & Szamuely (2006) p.105〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Central simple algebra」の詳細全文を読む スポンサード リンク
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