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. 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 4-dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals (see below).
| Attributes | Values |
|---|
| rdf:type
| |
| sameAs
| |
| foaf:isPrimaryTopicOf
| |
| rdfs:comment
| - 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. 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 4-dimensional CSA over R, and in fact represent the only non-trivial 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 finite-dimensional 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 4-dimensional CSA over R, and in fact represent the only non-trivial 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 finite-dimensional associative K-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 4-dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals (see below).
|
| rdfs:label
| |
| has abstract
| - 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. 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 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. 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 finite-dimensional 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 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. 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 finite-dimensional associative K-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 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. It is always a torsion group.
|
| Link to the Wikipage edit URL
| |
| Link from a Wikipage to an external page
| |
| extraction datetime
| |
| Link to the Wikipage history URL
| |
| Wikipage page ID
| |
| page length (characters) of wiki page
| |
| Wikipage modification datetime
| |
| Wiki page out degree
| |
| Wikipage revision ID
| |
| Link to the Wikipage revision URL
| |
| dbp:wikiPageUsesTemplate
| |
| dct:subject
| |
| is foaf:primaryTopic
of | |
| is rdfs:seeAlso
of | |
| is Wikipage disambiguates
of | |
| is Wikipage redirect
of | |
![[RDF Data]](/fct/images/sw-rdf-blue.png)
