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Lattice theoretic approach to space-time codes from division algebras
Related concepts (38)
Number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers).
Coding theory
Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studied by various scientific disciplines—such as information theory, electrical engineering, mathematics, linguistics, and computer science—for the purpose of designing efficient and reliable data transmission methods.
Simple ring
In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field. The center of a simple ring is necessarily a field. It follows that a simple ring is an associative algebra over this field. It is then called a simple algebra over this field. Several references (e.g., Lang (2002) or Bourbaki (2012)) require in addition that a simple ring be left or right Artinian (or equivalently semi-simple).
Frobenius theorem (real division algebras)
In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. According to the theorem, every such algebra is isomorphic to one of the following: R (the real numbers) C (the complex numbers) H (the quaternions). These algebras have real dimension 1, 2, and 4, respectively. Of these three algebras, R and C are commutative, but H is not.
Semisimple algebra
In ring theory, a branch of mathematics, a semisimple algebra is an associative artinian algebra over a field which has trivial Jacobson radical (only the zero element of the algebra is in the Jacobson radical). If the algebra is finite-dimensional this is equivalent to saying that it can be expressed as a Cartesian product of simple subalgebras. The Jacobson radical of an algebra over a field is the ideal consisting of all elements that annihilate every simple left-module.
Cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol . Given two linearly independent vectors a and b, the cross product, a × b (read "a cross b"), is a vector that is perpendicular to both a and b, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming.
Algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.
Octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface O or blackboard bold . Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative. They are also power associative.
Alternative algebra
In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have for all x and y in the algebra. Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as the octonions. Alternative algebras are so named because they are the algebras for which the associator is alternating. The associator is a trilinear map given by By definition, a multilinear map is alternating if it vanishes whenever two of its arguments are equal.
Division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element a has a multiplicative inverse, that is, an element usually denoted a^–1, such that a a^–1 = a^–1 a = 1. So, (right) division may be defined as a / b = a b–1, but this notation is avoided, as one may have a b^–1 ≠ b^–1 a. A commutative division ring is a field.