User gregingre - MathOverflow

Structure of units in a maximal order

2012-10-16T09:36:27Z

Hello,

my question is simple: do we have a "Dirichlet's unit theorem" for the group of units of a maximal order of a central division algebra ?

In other words: let $k$ be a number field, let $D$ be a central division $k$-algebra (i.e. a skew field with center $k$), and let $\Lambda$ be a maximal order over $\mathcal{O}_k$.

Is $\Lambda^\times$ a finitely generated group ? what is known about its group structure ?

I browsed the web and looked at Reiner's "Maximal orders" but didn't find anything.

I'm happy to assume that $D$ satisfies Eichler's condition if necessary.

In fact, my original question is even more precise: $k/\mathbb{Q}$ is quadratic imaginary, $D$ carries a unitary involution $\tau$ (which therefore restricts to complex conjugation on $k$), and I am interested in the structure of UNITARY units in a maximal order $\Lambda$.

If anyone knows any results/references, I would be happy to know them.

Thanks in advance!

Greg

Algebraic integers in skew fields

2012-10-12T15:35:54Z

Hi everyone,

let $D$ be a skew field, which is finite dimensional over its center $k$. Assume that $k$ is a number field, and let $\mathcal{O}_D$ be the set of elements $z\in D$ which are roots of a monic polynomial with coefficients in $\mathcal{O}_k$ . Is $\mathcal{O}_D$ a subring of $D$ ?

Thanks!

Comment by GreginGre

2012-10-16T12:47:32Z

thanks for your answers. I will try to look in the direction of arithmetic groups, then.

Comment by GreginGre

2012-10-16T09:38:49Z

thanks a lot! that was a silly question, especially because I already had this kind of counterexamples in the split case before posting my question ^^...