How to distinguish division algebras from matrix algebras? 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.
 A: 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.)
A: Tim, this is in general difficult (as the suggestions show). What I would do in/with Magma is
 - compute an explicit co-cycle parametrizing your central simple algebra. For this to work, you need a matrix representation over a number field that is normal over the centre of the algebra. (Basically compute the Galois action as conjugating matrices)
 - from the co-cycle you can compute the Schur-indices as the exponent in the Brauer group (either globally or locally)
 - comparing dimensions you should be able to see if you have a division algebra (skew-field) or not.
The MeatAxe approach can also be useful - Allan Steel developed this systematically further. However, in difficult cases, currently, one needs to revert to the Galois cohomology as above.
PS.: an improvement, while using the same core idea is to compute a maximal order of the algebra. The discriminant gives then enough information to derive the Schur indices, thus avoiding the Galois cohomology.
A: Let $k$ be the base field, and $K$ be a splitting field of your algebra. Then you can calculate the relative Brauer group as the second cohomology of the (simplicial) Amitsur complex $k^\times \to K^\times \to  K\otimes_k K^\times \to K\otimes_k K\otimes_k K^\times \to\dots$. Now, suppose that you know how to associate an element of  $M\in K\otimes_k K^\times$ to your algebra. Then it is easy to determine whether its splits (i.e. whether $M$ equals $x\otimes x^{-1}$ for some $x\in K^\times$);  in order to recover such an $x$ it suffices to project $K\otimes_k K$ onto $K$ by the linear map $a\otimes b\mapsto a\cdot  tr_{K/k}(cb)$ for some (fixed) $c\in K^{\times}$.
