 A quaternion algebra has a basis formula_4 where formula_5 and formula_6.A quaternion algebra is said to be split over formula_2 if it is isomorphic as an formula_2-algebra to the algebra of matrices formula_9; a quaternion algebra over an algebraically closed field is always split.If formula_10 is an embedding of formula_2 into a field formula_12 we shall denote by formula_13 the algebra obtained by extending scalars from formula_2 to formula_12 where we view formula_2 as a subfield of formula_12 via formula_10.