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Line 5: ==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 <math>\{ 1, i, j, k\}</math>, with the following multiplication rules: :<math>i^2=a</math> :<math>j^2=b</math> :<math>ij=k</math> :<math>ji=-k</math> where ''a'' and ''b'' are any given nonzero elements of ''F''. From these rules we get: ~~:''i''<sup>2</sup> = ''a''~~ :<math>k^2=ijij=-iijj=-ab</math> ~~:''j''<sup>2</sup> = ''b''~~ ~~:''ij'' = ''k''~~ ~~:''ji'' = −''k''~~ ~~where ''a'' and ''b'' are any given nonzero elements of ''F''. A short calculation shows ''k''<sup>2</sup> = −''ab''.~~ (The Hamilton quaternions are the case where <math>F=\mathbb{R}</math> and ''a'' = ''b'' = −1.) The algebra defined in this way is denoted (''a'',''b'')<sub>''F''</sub> or simply (''a'',''b'').<ref name=GS2>Gille & Szamuely (2006) p.2</ref> 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'')<sub>''F''</sub> is either a [[division algebra]] or isomorphic to the [[matrix algebra]] of 2×2 matrices over ''F'': the latter case is termed ''split''.<ref name=GS3>Gille & Szamuely (2006) p.3</ref> The ''norm form''

Quaternion algebra: Difference between revisions