How to distinguish division algebras from matrix algebras? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T21:54:22Z http://mathoverflow.net/feeds/question/57469 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57469/how-to-distinguish-division-algebras-from-matrix-algebras How to distinguish division algebras from matrix algebras? Tim Dokchitser 2011-03-05T16:43:20Z 2011-03-10T12:42:39Z <p>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?</p> <p>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.</p> http://mathoverflow.net/questions/57469/how-to-distinguish-division-algebras-from-matrix-algebras/57512#57512 Answer by Mikhail Bondarko for How to distinguish division algebras from matrix algebras? Mikhail Bondarko 2011-03-05T23:16:29Z 2011-03-05T23:16:29Z <p>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) $c\in K^{\times}$.</p> http://mathoverflow.net/questions/57469/how-to-distinguish-division-algebras-from-matrix-algebras/57757#57757 Answer by Claus Fieker for How to distinguish division algebras from matrix algebras? Claus Fieker 2011-03-08T01:03:22Z 2011-03-10T01:01:49Z <p>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.</p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/57469/how-to-distinguish-division-algebras-from-matrix-algebras/58065#58065 Answer by Henri Johnston for How to distinguish division algebras from matrix algebras? Henri Johnston 2011-03-10T12:42:39Z 2011-03-10T12:42:39Z <p>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? <a href="http://dx.doi.org/10.1016/j.jalgebra.2009.04.026" rel="nofollow">http://dx.doi.org/10.1016/j.jalgebra.2009.04.026</a></p> <p>Preprint version and magma code available here: <a href="http://www.math.rwth-aachen.de/~nebe/pl.html" rel="nofollow">http://www.math.rwth-aachen.de/~nebe/pl.html</a></p> <p>(I know this should really be a comment, but I'm afraid that I don't have enough reputation yet.)</p>