Suppose $D$ is an explicitly given rank 9 central simple algebra over ${\mathbb Q}$ (or a number field). For example it could be specified by two cubic subfields ${\mathbb Q}(a), {\mathbb Q}(b)\subset D$ and one relation $ba=\sum a\_i b\_j$. What is the best way to determine whether $D$ is a matrix algebra or a division algebra?


Any ideas or references would be highly appreciated! I simply tried to look for random elements whose minimal polynomial is reducible (sort of like Parker's meataxe for group representations), but this is not terribly efficient, and I don't know how to do this cleverly. Perhaps one could also take many cubic subfields and record which primes split in them - if the union of split primes over all subfields seems to cover all prime numbers, this suggests that $D$ is a matrix algebra; again, I don't know how to make this into an actual algorithm.



How to distinguish division algebras from matrix algebras?
This may be repeating what others have said as it essentially follows the maximal order approach, but have you looked at Nebe, Gabriele; Steel, Allan, Recognition of division algebras, J. Algebra 322 (2009), no. 3, 903–909?
<http://dx.doi.org/10.1016/j.jalgebra.2009.04.026>


Preprint version and magma code available here:
<http://www.math.rwth-aachen.de/~nebe/pl.html>


(I know this should really be a comment, but I'm afraid that I don't have enough reputation yet.)


