Matrix algebras over division algebras

[JRSijsling]

This is for later generalization in genus 4. How do we find out whether a given central simple algebra is, say, M_2 (B), where D is a quaternion algebra, instead of M_4 (QQ)?

For sure there are some approaches, like considering the discriminant of a maximal order. But is there a clean way? I know of none so far.

(Also think of stuff like matrix algebras over quaternion algebras over totally real fields with trivial finite discriminant...)

[jvoight]

If you want just to recognize, you use norm residue symbols; much of this is in magma (but a bit buried).

If you want to compute an isomorphism, there is the beginning of such an algorithm (https://www.sciencedirect.com/science/article/pii/S0021869312000300). I do not think it would be hard to generalize to an arbitrary central simple algebra, giving M_n(D) where D is a division ring, if that has not been done already.

[JRSijsling]

The last paragraph of the article cited does not involve any restrictions, as far as I can see, so if we suspect some isomorphism A ~ M_2 (B) with B a quaternion algebra, then we can just apply their methods. Or at least I hope so.

Yeah, I do want that isomorphism. Usually I want to be able to describe the order in the quaternion algebra as well, especially if it is Eichler say.

Anyway, also for the above we need some conjecture on the power above, so as to what B is. Can you tell me a little more about which norm residue symbols to use here? In the case, for example, where I have an abstract algebra A that I suspect to be isomorphic to some M_2 (B) with B a quaternion algebra.

[jvoight]

Cool!

Magma functionality is rudimentary, so the basic answer to your question is: pieces are there, it's likely a large task to get this to work in general, but a small to medium task to get it to work in an individual example. In other words, if we're not just talking about a few examples, we need to reboot the conversation.

To apply Galois cohomology in the nicest possible way, you need to write your algebra as a cyclic algebra. Enumerating elements you can find someone in your algebra of dimension 16 whose reduced charpoly is irreducible generating a cyclic extension K. Usually "small" elements suffice.

After that, you write your algebra as a cyclic algebra: write your algebra as a sided vector space of dimension 4 over K and search for a complementary cyclic generator.

After it's cyclic, now you can apply norm residue symbols. Look in the documentation for "IsGloballySplit", and I wish you good luck.

If you don't want to think this hard, you can guess the quaternion algebra B0 and the decomposition B = M_2(B0) as follows. Compute the charpoly of any element alpha in B; it is always a 4th power, so take the 4th root to get the reduced charpoly. 100% of the time, you should see an irreducible such polynomial c(alpha;T). Let p be prime, and consider the factorization of c(alpha;T) over ZZ_p. If you ever see a polynomial c(alpha;T) which is separable and splits completely over ZZ_p (so a product of 4 distinct linear factors), then you have proven that B_p = M_4(QQ_p). I think you can get the idea from here about how to get more refined things, but please ask if this is a line you'd like to pursue further.

[JRSijsling]

I am fine with "just" doing genus 4 examples, after that (as g grows, that is) nothing essentially new shows up for a while.

And yeah, in that genus I want something general. But if I can guess with one part and verify with the other, and that works in practice, then I am happy. Will try that... at some point.

[JRSijsling]

This (and finding decent idempotents in this situation if they exist) can be work for a good master's student and is no priority. Closing.