caveman

@loldop, well it's related to the gaussian integral in some way... not sure how

wrhats wrong

bye

an automorphism of the ring of 2x2 integer matrices

then is the set of Z[sqrt(5)] representations closed under this automorphism?

but can we extend it to the whole ring of 2x2 integer matrices?

so there is easily an automorphism from one to another

they're all copies of the same ring

I mean

ok

will look it up

I dontknow about canonical algebras

conjugation automorphisms, but also automorphisms that permute the subrings

yes @awllower, but similar to your remark about 5 being small: it is clear that ther are always finitely may solutions to the equation

is there some structure on the set of these rings?

it is very curious to me

why are there many of them?

I am interesting in what these rings isomorphic to Z[sqrt(5)] are

so what are the solutions of alpha^2 + beta gamma = 5? I mean that's not Pell equation

I multiplied my matrix wrong

that was stupid of me

oh I see, sorry

yes

where is that equation from?

I don't see why you have alpha + delta = 0

@awllower, yes that diophantine equation-

@awllower, I feel like I could Halls theorem eexcept a the last (most difficult) case

sorry awllower that message was for Ethan

@Lord_Farin, I deleted it

just write "let X be a complete residue system mod a"

@awllower, regarding the copies of Z[sqrt(5)]

well, my stuff is all about the finite field matrix groups

my understanding of matrix groups isn't really good enough..

trying to do some more group theory today but I can't decide what

I see

seems like all this stuff matters in algebraic geometry which I have no idea about

I don't know the importance of a monadic functor

i can verify a proof but I just feel like "what the heck is going on"

dont know if you've encountered it

@Lord_Farin, monadicity is absolutely ridiculous though

the proof is a lot of fun, you just follow your nose and a commutative prism pops out at the end

If you have an initial object in $A \downarrow G$ for every A, then G has a left adjoint

the downarrow theorem is really nice though

there is yet another adjoint functor theorem where you have the assumption that something is "well powered"

and from the weakly initial set you can build an initial object with a little trick then you get the adjoint functor by a general theorem about downarrow categories

that's it

I have another adjoint functor theorem: ummmm.. let C have small limits, F preserve limits and have a weakly initial set in each downarrow category I think

I suppose it still has some use

seems like a useful theorem until you see that the assumptions force the category to be a poset

it came up for me from Freyd's adjoint function theorem