Quaternion algebra

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a quaternion algebra over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> F is a <a href="/facts/Central_simple_algebra/RsBMUkIW">central simple algebra</a> A over F that has <a href="/facts/Dimension_(vector_space)/z8m02A3W">dimension</a> 4 over F. Every quaternion algebra becomes a <a href="/facts/Matrix_algebra/oj5HBr8B">matrix algebra</a> by <a href="/facts/Scalar_extension/RYGsTsYk">extending scalars</a> (equivalently, <a href="/facts/Tensor_product_of_algebras/TdWszpzf">tensoring</a> with a <a href="/facts/Field_extension/L7yIDtyK">field extension</a>), i.e. for a suitable field extension K of F, 
 
 
 
 A
 
 ⊗
 
 F
 
 
 K
 
 
 {\displaystyle A\otimes _{F}K}
 
 is <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> to the 2 × 2 matrix algebra over K.
The notion of a quaternion algebra can be seen as a generalization of Hamilton's <a href="/facts/Quaternion/T3tnu7mX">quaternions</a> to an arbitrary base field. The Hamilton quaternions are a quaternion algebra (in the above sense) over 
 
 
 
 F
 =
 
 R
 
 
 
 {\displaystyle F=\mathbb {R} }
 
, and indeed the only one over 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 apart from the 2 × 2 <a href="/facts/Real_number/R02gw5Pb">real</a> matrix algebra, up to isomorphism. When 
 
 
 
 F
 =
 
 C
 
 
 
 {\displaystyle F=\mathbb {C} }
 
, then the <a href="/facts/Biquaternion/YsI2czSu">biquaternions</a> form the quaternion algebra over F.

Quaternion algebra open-in-new

Quaternion algebra