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).

CSAs over a field *K* are a non-commutative analog to extension fields over *K* – 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 a division algebra). This is of particular interest in noncommutative number theory as generalizations of number fields (extensions of the rationals **Q**); see noncommutative number field.

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*.

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.

We call a field *E* a *splitting field* for *A* if *A*⊗*E* is isomorphic to a matrix ring over *E*. Every finite dimensional CSA has a splitting field: indeed, in the case when *A* is a division algebra, then a maximal subfield of *A* is a splitting field. As an example, the field **C** splits the quaternion algebra **H** over **R** with

We can use the existence of the splitting field to define **reduced norm** and **reduced trace** for a CSA *A*. Map *A* 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 algebra **H**, the element *t* + *x* **i** + *y* **j** + *z* **k** has reduced norm *t*2 + *x*2 + *y*2 + *z*2 and reduced trace 2*t*.

Read more about Central Simple Algebra: Properties

### Other articles related to "simple, algebra, central simple algebra, simple algebra":

... The

**simple**past or past

**simple**, sometimes called the preterite, is the basic form of the past tense in Modern English ... The term "

**simple**" is used to distinguish the syntactical construction whose basic form uses the plain past tense alone, from other past tense constructions which ... Regular verbs form the

**simple**past in -ed however there are a few hundred irregular verbs with different forms ...

... of order n can be represented by a cyclic division

**algebra**of dimension n2 ... If D is a

**central simple algebra**over K and v is a valuation then D ⊗ Kv is a

**central simple algebra**over Kv, the local completion of K at v ... A given

**central simple algebra**D splits for all but finitely many v, so that the image of D under almost all such homomorphisms is 0 ...

... Every finite-dimensional

**simple algebra**over R must be a matrix ring over R, C, or H ... Every

**central simple algebra**over R must be a matrix ring over R or H ... Every finite-dimensional

**simple algebra**over C must be a matrix ring over C and hence every

**central simple algebra**over C must be a matrix ring over C ...

**Central Simple Algebra**- Properties

... There is a unique division

**algebra**in each Brauer equivalence class ... 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 ...

### Famous quotes containing the words algebra, central and/or simple:

“Poetry has become the higher *algebra* of metaphors.”

—José Ortega Y Gasset (1883–1955)

“In inner-party politics, these methods lead, as we shall yet see, to this: the party organization substitutes itself for the party, the *central* committee substitutes itself for the organization, and, finally, a “dictator” substitutes himself for the *central* committee.”

—Leon Trotsky (1879–1940)

“He prayed more deeply for *simple* selflessness than he had ever prayed before—and, feeling an uprush of grace in the very intention, shed the night in his heart and called it light. And walking out of the little church he felt confirmed in not only the worth of his whispered prayer but in the realization, as well, that Christ had become man and not some bell-shaped Corinthian column with volutes for veins and a mandala of stone foliage for a heart.”

—Alexander Theroux (b. 1940)