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==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:
:<math>k^2=ijij=-iijj=-ab</math>
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''
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