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I've developed a framework in Maple for playing with cyclic division algebras. I have methods for multiplication and passing to and from:

The standard basis over the base field in terms of x^i*y^j and:
The standard matrix basis for the central simple algebra when taking coefficients in the splitting field.

I have it done in the case of degree 7, but could generalise it to any prime degree if this is of use to anyone.
algebra cyclic

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