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In [[mathematics]], a '''quaternion algebra''' over a field ''F'' is a [[central simple algebra]] ''A'' over ''F''<ref>See Peirce. Associative algebras. Springer. Lemma at page 14.</ref><ref>See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2.</ref> that has dimension 4 over ''F''. Every quaternion algebra becomes the matrix algebra by ''extending [[scalar (mathematics)|scalars]]'' (=[[tensor product of algebras|tensoring]] with a field extension), i.e. for a suitable [[field extension]] ''K'' of ''F'', <math>A \otimes_F K</math> is isomorphic to the 2×2 [[matrix algebra]] over ''K''.
The notion of a quaternion algebra can be seen as a generalization of the [[Hamilton quaternions]] to an arbitrary [[base field]]. The Hamilton quaternions are a quaternion algebra (in the above sense) over <math>F = \mathbb{R}</math> (the [[real number field]]), and indeed the only one over <math>\mathbb{R}</math> apart from the 2×2 [[real matrix]] algebra, up to [[isomorphism]]. When <math>F = \mathbb{C}</math>, then the [[biquaternion]]s form the quaternion algebra over ''F''.
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