• mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible...
    8 KB (1,153 words) - 17:39, 1 May 2024
  • Thumbnail for Division (mathematics)
    left-distributive and right-distributive, and thus distributive. Division is often shown in algebra and science by placing the dividend over the divisor with...
    25 KB (3,478 words) - 16:38, 15 May 2025
  • Thumbnail for Quaternion
    Quaternion (category Composition algebras)
    normed division algebra over the real numbers, and therefore a ring, also a division ring and a domain. It is a special case of a Clifford algebra, classified...
    96 KB (12,665 words) - 22:22, 26 May 2025
  • abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over...
    10 KB (1,280 words) - 22:13, 19 November 2024
  • mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure...
    65 KB (9,287 words) - 07:33, 12 May 2025
  • quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes a matrix algebra by extending...
    11 KB (1,579 words) - 21:29, 14 May 2025
  • is called a Euclidean Hurwitz algebra or (finite-dimensional) normed division algebra. If A is a Euclidean Hurwitz algebra and a is in A, define the involution...
    28 KB (3,682 words) - 00:14, 19 May 2025
  • mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A {\displaystyle A} over the real or complex...
    17 KB (2,618 words) - 09:10, 24 May 2025
  • In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center...
    31 KB (4,261 words) - 10:53, 26 May 2025
  • In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined...
    12 KB (1,482 words) - 06:05, 20 February 2025
  • N(x)=xx^{*}} is called the norm of the algebra. A composition algebra (A, ∗, N) is either a division algebra or a split algebra, depending on the existence of...
    11 KB (1,319 words) - 14:21, 10 October 2024
  • 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...
    8 KB (1,140 words) - 04:54, 10 December 2024
  • Thumbnail for Projective space
    definition, which is more often encountered in modern textbooks. Using linear algebra, a projective space of dimension n is defined as the set of the vector...
    37 KB (5,670 words) - 20:15, 2 March 2025
  • numbers C. The octonion algebra for N is a division algebra if and only if the form N is anisotropic. A split octonion algebra is one for which the quadratic...
    7 KB (820 words) - 15:34, 20 February 2025
  • A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative...
    25 KB (3,005 words) - 20:16, 18 February 2025
  • Brauer group (category Algebraic number theory)
    to K. Note that CSAs are in general not division algebras, though CSAs can be used to classify division algebras. For example, the complex numbers C form...
    22 KB (2,937 words) - 18:11, 30 April 2025
  • Thumbnail for Abstract algebra
    In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations...
    33 KB (4,336 words) - 09:19, 28 April 2025
  • In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have x...
    7 KB (1,080 words) - 23:15, 17 May 2025
  • Octonion (redirect from Dixon algebra)
    In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented...
    42 KB (5,316 words) - 02:52, 26 February 2025
  • Simple ring (redirect from Simple algebra)
    finite-dimensional simple algebra over a field k {\displaystyle k} , it is isomorphic to a matrix algebra over some division algebra over k {\displaystyle...
    6 KB (852 words) - 23:01, 23 March 2025
  • for algebras over a field k. If R is a finite-dimensional semisimple k-algebra, then each Di in the above statement is a finite-dimensional division algebra...
    8 KB (1,085 words) - 00:57, 5 May 2024
  • In mathematics, an Azumaya algebra is a generalization of central simple algebras to R {\displaystyle R} -algebras where R {\displaystyle R} need not...
    17 KB (3,208 words) - 22:29, 28 October 2023
  • universal algebra, an algebraic structure is called an algebra; this term may be ambiguous, since, in other contexts, an algebra is an algebraic structure...
    21 KB (2,706 words) - 16:56, 23 May 2025
  • of a quaternionic vector space (i.e., V becomes a module over the division algebra of quaternions). From this point of view, quaternionic representation...
    5 KB (689 words) - 13:57, 25 May 2025
  • mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables...
    75 KB (9,572 words) - 09:14, 22 April 2025
  • Thumbnail for Emmy Noether
    German mathematician who made many important contributions to abstract algebra. She also proved Noether's first and second theorems, which are fundamental...
    134 KB (15,260 words) - 12:03, 18 May 2025
  • idea independently. It associates a Lie algebra to a pair of division algebras A, B. The resulting Lie algebras have Dynkin diagrams according to the table...
    29 KB (2,914 words) - 21:30, 15 May 2024
  • Thumbnail for Field (mathematics)
    multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental algebraic structure...
    87 KB (10,305 words) - 10:23, 28 May 2025
  • {O} } , and the Frobenius theorem says the only real associative division algebras are R {\displaystyle \mathbb {R} } , C {\displaystyle \mathbb {C}...
    27 KB (3,215 words) - 12:59, 17 May 2025
  • composition algebras frequently applied in mathematical physics. The Cayley–Dickson construction defines a new algebra as a Cartesian product of an algebra with...
    21 KB (2,554 words) - 06:23, 7 May 2025