User gregingre - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T00:35:49Z http://mathoverflow.net/feeds/user/27189 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109800/structure-of-units-in-a-maximal-order Structure of units in a maximal order GreginGre 2012-10-16T09:36:27Z 2012-10-18T10:55:03Z <p>Hello,</p> <p>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 ?</p> <p>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 $\mathcal{O}_k$.</p> <p>Is $\Lambda^\times$ a finitely generated group ? what is known about its group structure ?</p> <p>I browsed the web and looked at Reiner's "Maximal orders" but didn't find anything.</p> <p>I'm happy to assume that $D$ satisfies Eichler's condition if necessary.</p> <p>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 $\Lambda$.</p> <p>If anyone knows any results/references, I would be happy to know them.</p> <p>Thanks in advance!</p> <p>Greg </p> http://mathoverflow.net/questions/109464/algebraic-integers-in-skew-fields Algebraic integers in skew fields GreginGre 2012-10-12T15:35:54Z 2012-10-12T16:23:04Z <p>Hi everyone,</p> <p>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 $D$ ?</p> <p>Thanks!</p> <p>G.</p> http://mathoverflow.net/questions/109800/structure-of-units-in-a-maximal-order Comment by GreginGre GreginGre 2012-10-16T12:47:32Z 2012-10-16T12:47:32Z thanks for your answers. I will try to look in the direction of arithmetic groups, then. http://mathoverflow.net/questions/109464/algebraic-integers-in-skew-fields/109467#109467 Comment by GreginGre GreginGre 2012-10-16T09:38:49Z 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 ^^...