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In mathematics, a quaternion algebra over a field F is a central simple algebra A over F [1] [2] that has dimension 4 over F.Every quaternion algebra becomes a matrix algebra by extending scalars (equivalently, tensoring with a field extension), i.e. for a suitable field extension K of F, is isomorphic to the 2 × 2 matrix algebra over K.
The Quaternions can be generalized into further algebras called quaternion algebras. Take F to be any field with characteristic different from 2, and a and b to be elements of F; a four-dimensional unitary associative algebra can be defined over F with basis 1, i, j, and i j, where i 2 = a, j 2 = b and i j = −j i (so (i j) 2 = −a b).
In mathematics, an Eichler order, named after Martin Eichler, is an order of a quaternion algebra that is the intersection of two maximal orders. References [ edit ]
For n = 2 it is an associative algebra called a quaternion algebra, and for n = 3 it is an alternative algebra called an octonion algebra. These instances n = 1, 2 and 3 produce composition algebras as shown below. The case n = 1 starts with elements (a, b) in F × F and defines the conjugate (a, b)* to be (a*, –b) where a* = a in case n = 1 ...
In group theory, the quaternion group Q 8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset {,,,,,} of the quaternions under multiplication. It is given by the group presentation
The first part explains the Cayley–Dickson construction, [1] [3] which constructs the complex numbers from the real numbers, the quaternions from the complex numbers, and the octonions from the quaternions. Related algebras are also discussed, including the sedenions (a 16-dimensional real algebra formed in the same way by taking one more ...
A quaternion algebra over a field is a four-dimensional central simple-algebra. A quaternion algebra has a basis ,,, where , and =. A quaternion algebra is said to be split over if it is isomorphic as an -algebra to the algebra of matrices ().
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, while any matrix ring over the reals or quaternions – M(n, R) or M(n, H) – is a CSA over the reals, but not a division algebra (if n > 1).