Quaternion Algebras

A quaternion algebra is a central, simple algebra of dimension four over a field. A special type for quaternion algebras is released in Magma V2.7. Support for orders over $\mbox{\bf Z}$ and k[x] and their ideals is provided for quaternions over the rational field $\mbox{\bf Q}$ or k(x). Special functions for enumeration all ideals in definite quaternion algebras over $\mbox{\bf Q}$, with connections to modular forms.

• Arithmetic of elements
• Norm, trace, and conjugation
• Minimal polynomial of elements
• Discriminant and ramified primes
• Creation of prime ideals
• Testing for principal ideals
• Enumeration of left and right ideals of an definite order over $\mbox{\bf Z}$
• Left and right orders of an ideal in a definite order over $\mbox{\bf Z}$