Quaternion algebra
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In mathematics, a quaternion algebra over a field F is a particular kind of central simple algebra A over F, namely such an algebra that has dimension 4, and therefore becomes the 2×2 matrix algebra over some field extension of F, by extending scalars. The classical quaternions are the case of F the real number field, and A is uniquely defined up to isomorphism by the condition that it is such a quaternion algebra that is not the 2×2 real matrix algebra.
Quaternion algebra therefore means something other than the algebra of quaternions. In fact when the base field F does not have characteristic 2, any quaternion algebra over F is a slightly twisted form of the familiar quaternions, with a basis 1, i, j, and k such that
- i2 = a
- j2 = b
- ij = k, ji = −k
where a and b are nonzero elements of F and need not be −1. When F has characteristic 2, a different explicit description in terms of a basis of 4 elements is also possible, but in any event the description as a 4-dimensional central simple algebra applies uniformly in all characteristics.
Quaternion algebras are applied in number theory, particularly to quadratic forms. They are concrete structures that represent the elements of order two in the Brauer group. In particular, for p-adic fields, their theory includes that of the Hilbert symbol of local class field theory.
