Inring theory and related areas ofmathematics acentral simple algebra (CSA) over afieldK is a finite-dimensionalassociativeK-algebraA that issimple, and for which thecenter is exactlyK.

For example, thecomplex numbersC form a CSA over themselves, but not over thereal numbersR (the center ofC is all ofC, not justR). ThequaternionsH form a 4-dimensional CSA overR, and in fact represent the only non-trivial element of theBrauer group of the reals (see below). Finite-dimensionality is essential to the definition: for instance, for a fieldF of characteristic 0, theWeyl algebra${\displaystyle F[X,{\mathrm{\partial }}_X]}$ is a simple algebra with centerF, but isnot a central simple algebra overF as it has infinite dimension as aF-module.

By theArtin–Wedderburn theorem, a finite-dimensional simple algebraA is isomorphic to the matrix algebraM(n,S) for somedivision ringS. Given two central simple algebrasA ~M(n,S) andB ~M(m,T) over the same fieldF,A andB are calledsimilar (orBrauer equivalent) if their division ringsS andT are isomorphic. The set of allequivalence classes of central simple algebras over a given fieldF, under this equivalence relation, can be equipped with agroup operation given by thetensor product of algebras. The resulting group is called theBrauer group Br(F) of the fieldF.^[1] It is always atorsion group.^[2]

• According to theArtin–Wedderburn theorem a finite-dimensional simple algebraA is isomorphic to the matrix algebraM(n,S) for somedivision ringS. Hence, there is a unique division algebra in each Brauer equivalence class.^[3]
• Everyautomorphism of a central simple algebra is aninner automorphism (this follows from theSkolem–Noether theorem).
• Thedimension of a central simple algebra as a vector space over its centre is always a square: thedegree is the square root of this dimension.^[4] TheSchur index of a central simple algebra is the degree of the equivalent division algebra:^[5] it depends only on theBrauer class of the algebra.^[6]
• Theperiod orexponent 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,^[7] and the two numbers are composed of the same prime factors.^[8][9][10]
• IfS is a simplesubalgebra of a central simple algebraA then dim_F S divides dim_F A.
• Every 4-dimensional central simple algebra over a fieldF is isomorphic to aquaternion algebra; in fact, it is either a two-by-twomatrix algebra, or adivision algebra.
• IfD is a central division algebra overK for which the index has prime factorisation

${\displaystyle \mathrm{i}\mathrm{n}\mathrm{d}(D)=\prod _{i=1}^r{p}_i^{m_i}}$

thenD has a tensor product decomposition

${\displaystyle D=\underset{i=1}{\overset{r}{⨂}}{D}_i}$

where each componentD_i is a central division algebra of index${\displaystyle p_i^{m_i}}$, and the components are uniquely determined up to isomorphism.^[11]

We call a fieldE asplitting field forA overK ifA⊗E is isomorphic to a matrix ring overE. Every finite dimensional CSA has a splitting field: indeed, in the case whenA is a division algebra, then amaximal subfield ofA is a splitting field. In general by theorems ofWedderburn and Koethe there is a splitting field which is aseparable extension ofK of degree equal to the index ofA, and this splitting field is isomorphic to a subfield ofA.^[12][13] As an example, the fieldC splits the quaternion algebraH overR with

We can use the existence of the splitting field to definereduced norm andreduced trace for a CSAA.^[14] MapA to a matrix ring over a splitting field and define the reduced norm and trace to be the composite of this map with determinant and trace respectively. For example, in the quaternion algebraH, the splitting above shows that the elementt +xi +yj +zk has reduced normt² +x² +y² +z² and reduced trace 2t.

The reduced norm is multiplicative and the reduced trace is additive. An elementa ofA is invertible if and only if its reduced norm is non-zero: hence a CSA is a division algebra if and only if the reduced norm is non-zero on the non-zero elements.^[15]

CSAs over a fieldK are a non-commutative analog toextension fields overK – in both cases, they have no non-trivial 2-sided ideals, and have a distinguished field in their center, though a CSA can be non-commutative and need not have inverses (need not be adivision algebra). This is of particular interest innoncommutative number theory as generalizations ofnumber fields (extensions of the rationalsQ); seenoncommutative number field.

• Azumaya algebra, generalization of CSAs where the base field is replaced by a commutative local ring
• Severi–Brauer variety
• Posner's theorem

• Cohn, P.M. (2003).Further Algebra and Applications (2nd ed.). Springer.ISBN 1852336676.Zbl 1006.00001.
• Jacobson, Nathan (1996).Finite-dimensional division algebras over fields. Berlin:Springer-Verlag.ISBN 3-540-57029-2.Zbl 0874.16002.
• Lam, Tsit-Yuen (2005).Introduction to Quadratic Forms over Fields.Graduate Studies in Mathematics. Vol. 67. American Mathematical Society.ISBN 0-8218-1095-2.MR 2104929.Zbl 1068.11023.
• Lorenz, Falko (2008).Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer.ISBN 978-0-387-72487-4.Zbl 1130.12001.

• Albert, A.A. (1939).Structure of Algebras. Colloquium Publications. Vol. 24 (7th revised reprint ed.). American Mathematical Society.ISBN 0-8218-1024-3.Zbl 0023.19901.{{cite book}}:ISBN / Date incompatibility (help)
• Gille, Philippe; Szamuely, Tamás (2006).Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics. Vol. 101. Cambridge:Cambridge University Press.ISBN 0-521-86103-9.Zbl 1137.12001.

• Algebras
• Ring theory

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