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.

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.