Quaternion Algebra - Structure

Structure

Quaternion algebra here means something more general than the algebra of Hamilton quaternions. When the coefficient field F does not have characteristic 2, every quaternion algebra over F can be described as a 4-dimensional F-vector space with basis, with the following multiplication rules:

i2 = a

j2 = b

ij = k

ji = −k

where a and b are any given nonzero elements of F. A short calculation shows k2 = −ab. (The Hamilton quaternions are the case where and a = b = −1.) The algebra defined in this way is denoted (a,b)_F or simply (a,b). 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 definition of a quaternion algebra over F as a 4-dimensional central simple algebra over F applies uniformly in all characteristics.

A quaternion algebra (a,b)_F is either a division algebra or isomorphic to the matrix algebra of 2×2 matrices over F: the latter case is termed split. The norm form

defines a structure of division algebra if and only if the norm is an anisotropic quadratic form, that is, zero only on the zero element. The conic C(a,b) defined by

has a point (x,y,z) with coordinates in F in the split case.