Azumaya algebra & Field extension - Unionpedia, the concept map

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Difference between Azumaya algebra and Field extension

Azumaya algebra vs. Field extension

In mathematics, an Azumaya algebra is a generalization of central simple algebras to R-algebras where R need not be a field. In mathematics, particularly in algebra, a field extension (denoted L/K) is a pair of fields K \subseteq L, such that the operations of K are those of L restricted to K. In this case, L is an extension field of K and K is a subfield of L. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.

Algebraic geometry

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems.

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Brauer group

In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras over K, with addition given by the tensor product of algebras.

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Center (ring theory)

In algebra, the center of a ring R is the subring consisting of the elements x such that for all elements y in R. It is a commutative ring and is denoted as Z(R); 'Z' stands for the German word Zentrum, meaning "center".

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Central simple algebra

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. (Note that not every simple algebra is a central simple algebra over its center: for instance, if K is a field of characteristic 0, then the Weyl algebra K is a simple algebra with center K, but is not a central simple algebra over K as it has infinite dimension as a K-module.) 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).

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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers.

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Local ring

In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.

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Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

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P-adic number

In number theory, given a prime number, the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right.

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