1 Algebraic view
Let be a field. The Brauer group of is the set of all equivalence classes of central simple algebras over , where two central simple algebras and are equivalent if there exists a division ring over and natural numbers such that (resp. ) is isomorphic to the ring of (resp. ) matrices with coefficients in .
The group operation in is given by tensor product: for any two central simple algebras over , their product in is the central simple algebra . The identity element in is the class of itself, and the inverse of a central simple algebra is the opposite algebra defined by reversing the order of the multiplication operation of .
2 Cohomological view
The Brauer group of is naturally isomorphic to the second Galois cohomology group . See http://www.math.harvard.edu/ elkies/M250.01/index.htmlhttp://www.math.harvard.edu/ elkies/M250.01/index.html Theorem 12 and succeeding remarks.
|Date of creation||2013-03-22 11:49:31|
|Last modified on||2013-03-22 11:49:31|
|Last modified by||djao (24)|