Introduction

A quaternion algebra A over a field K is a central simple algebra of dimension four over K. The matrix algebra M_2(K), the split quaternion algebra, is a trivial case. Every quaternion algebra over K not isomorphic to M_2(K) is a division algebra. Over some quadratic field extension L/K, the extension of scalars to L is the matrix algebra M_2(L). The latter fact makes the inclusion of split quaternion algebras important for the purpose of having a well-defined class of algebras which are preserved under base extension. In particular, we note that a quaternion algebra can be formed over a finite field K, where every quaternion algebra is isomorphic to a matrix algebra. A quaternion algebra in Magma is a specialized type of dimension 4 associative algebra over a field, which has the type AlgQuat.

A principal interest in the study of quaternion algebras is the orders and ideals over an integral domain R with quotient field K. Specialized functions are available for quaternion algebras over the rational field Q. Facilities for enumeration of all isomorphism classes of left or right ideals over an order are implemented, and this ties into the machinery for constructing the Brandt module -- a free abelian group on these ideal classes, on which Brandt matrices act as the adjacency matrices of the graph of ideals under a certain p-neighbour relation on ideals. It is also possible to construct orders over a rational function field k(x), for a field k. Orders and ideals in Magma have a common type AlgQuatOrd and are distinguished only by the presence of a unity. The main reference for material in this chapter is the book of Vign'eras [Vig80].