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In mathematics, the Brauer group arose out of an attempt to classify division algebras over a given field K. It is an abelian group with elements isomorphism classes of division algebras over K, such that the center is exactly K. The group is named for the algebraist Richard Brauer. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division is possible. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ...
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...
The term center is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. ...
Richard Dagobert Brauer (February 10, 1901 - April 17, 1977) was a leading German and American mathematician. ...
Construction of the Brauer group
A central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A, which is a simple ring, and for which the center is exactly K. For example, the complex numbers C form a CSA over themselves, but not over R (the center is C itself, hence too large to be CSA over R). The finite-dimensional division algebras with center R (that means the dimension over R is finite) are the real numbers and the quaternions by a theorem of Frobenius. In ring theory and related areas of mathematics a central simple algebra (CSA) over K, also called a Brauer algebra after Richard Brauer, is a finite-dimensional (associative) algebra A, which is a simple ring, and for which the center is exactly K. For example, the complex numbers C form...
In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ...
In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. ...
The term center is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. ...
A picture of Frobenius Ferdinand Georg Frobenius (October 26, 1849 – August 3, 1917) was a German mathematician, best-known for his contributions to the theory of differential equations and to group theory. ...
Given central simple algebras A and B, one can look at the their tensor product A ⊗ B as a K-algebra (see tensor product of R-algebras). It turns out that this is always central simple. A slick way to see this is to use a characterisation: a central simple algebra over K is a K-algebra that becomes a matrix ring when we extend the field of scalars to an algebraic closure of K. In mathematics, there is a construction in abstract algebra of the tensor product of commutative rings; which puts a ring structure on the tensor product as abelian groups of two commutative rings R and S. This structure on RZS then makes it a coproduct in the category of commutative rings. ...
In abstract algebra the matrix ring M(n,R) is set of all n-by-n matrices over an arbitrary ring R. This forms a ring under matrix addition and multiplication. ...
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ...
Given this closure property for CSAs, they form a monoid under tensor product. To get a group, apply the Artin-Wedderburn theorem (Wedderburn's part, in fact), to express any CSA as a M(n,D) for some division algebra D. If we look just at D, rather than the value of n, the monoid becomes a group. That is, if we impose an equivalence relation identifying M(m,D) with M(n,D) for all integers m and n at least 1, we get an equivalence relation; and the equivalence classes are all invertible: the inverse class to that of an algebra A is the one containing the opposite algebra Aop (the opposite ring with the same action by K since the image of K → A is in the center of A). In other words, for a CSA A we have A ⊗ Aop = M(n2,F), where n is the degree of A over F. (This provides a substantial reason for caring about the notion of an opposite algebra: it provides the inverse in the Brauer group.) In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ...
In abstract algebra, the Artin-Wedderburn theorem is a classification theorem for semisimple product of ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely determined up to permutation of the index i. ...
Joseph Henry Maclagen Wedderburn (2 February 1882- 9 October 1948) was a Scottish mathematician, who from 1909 had positions at Princeton University. ...
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In mathematics, an equivalence relation is a binary relation between two elements of a set which groups them together as being equivalent in some way. ...
In mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. ...
In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...
The term center is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. ...
Examples The Brauer group for an algebraically closed field or a finite field is the trivial group with only the identity element. In mathematics, a field is said to be algebraically closed if every polynomial in one variable of degree at least , with coefficients in , has a zero (root) in . ...
In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ...
The following list in mathematics contains the finite groups of small order up to group isomorphism. ...
The Brauer group Br(R) of the real number field R is a cyclic group of order two: there are just two types of division algebras, R and the quaternion algebra H. The product in the Brauer group is based on the tensor product: the statement that H has order two in the group is equivalent to the existence of an isomorphism of R-algebras: H ⊗ H ≅ M(4,R), where the RHS is the ring of 4×4 real matrices. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
In group theory, a cyclic group or monogenous group is a group that can be generated by a single element, in the sense that the group has an element g (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of...
In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...
In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ...
In mathematics, LHS is informal shorthand for the left-hand side of an equation. ...
Tsen's theorem implies that the Brauer group of a function field in one variable over an algebraically closed field vanishes. In mathematics, Tsens theorem states that a function field K of an algebraic curve over an algebraically closed field is quasi-algebraically closed. ...
Further theory In the further theory, the Brauer group of a local field is computed (it turns out to be canonically isomorphic to Q/Z for any local field, of characteristic 0 or characteristic p) and the results are applied to global fields. This gives one approach to class field theory, which was the first approach that allowed global class field theory to be derived from local class field theory; historically it had been the other way around at first. It also has been applied to Diophantine equations. More precisely, the Brauer group Br(K) of a global field K is given by the exact sequence In mathematics, a local field is a special type of field which has the additional property that it is a complete metric space with respect to a discrete valuation. ...
The term global field refers to either of the following: a number field, i. ...
In mathematics, class field theory is a major branch of algebraic number theory. ...
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. ...
 where the direct sum in the middle is over all (archimedean and non-archimedean) completions of K and the map to  is addition, where we interpret the Brauer group of the reals as (1/2)Z/Z. The group Q/Z on the right is really the "Brauer group" of the class formation of idele classes associated to K. In mathematics, a class formation is a structure used to organize the various Galois groups and modules that appear in class field theory. ...
In the general theory the Brauer group is expressed by factor sets; and expressed in terms of Galois cohomology via In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. ...
 Here, not assuming K to be a perfect field, Ks is the separable closure. When K is perfect this is the same as an algebraic closure; otherwise the Galois group must be defined in terms of Ks/K even to make sense. In mathematics, a separable extension of a field K is a field L containing K that can be generated by adjoining to K a set of elements α, each of which is a root of a separable polynomial over K. In that case, each β in L has a separable...
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ...
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ...
A generalisation via the theory of Azumaya algebras was introduced in algebraic geometry by Grothendieck. In mathematics, an Azumaya algebra is a generalization of central simple algebras to R-algebras where R need not be a field. ...
Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ...
Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is one of the most important mathematicians of the 20th century. ...
See also In ring theory and related areas of mathematics a central simple algebra (CSA) over K, also called a Brauer algebra after Richard Brauer, is a finite-dimensional (associative) algebra A, which is a simple ring, and for which the center is exactly K. For example, the complex numbers C form...
In mathematics, a class formation is a structure used to organize the various Galois groups and modules that appear in class field theory. ...
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