A Theorem on the Unit Groups of Simple Algebras

There are essential arithmetical differences between algebras which satisfy the i?-condition (i?-algebras) and those which do not, especially with regard to class-number properties (Eichler [l, 2, 3]). The meaning of the i?-condition in the case n = 2 is as follows. Both k and A are simple algebras over the field ko of rational numbers, of orders m and 4m, respectively, over &oSuppose ko is extended to the field ki of real numbers. Then the extended algebra kXki is the direct sum of fields, each of which is isomorphic either to k\ or to the field #2 of complex numbers. This decomposition of kXk\ involves the decomposition of A Xki into a direct sum of simple algebras over ki, the centers of which are the corresponding summands of kXki. Each summand of A Xk\ is either (1) a matrix algebra of degree 2 over k±, (2) a matrix algebra of degree 2 over k%, or (3) the division algebra of quaternions over k\. With each summand of kXki is associated an infinite prime place of k which is said to be ramified or unramified in A according as the corresponding summand of A Xk% is (3) or is either (1) or (2). The i?-condition for w = 2 is thus equivalent to requiring that not all summands of AXh be (3), in other words, that A over k is not a totally definite quaternion algebra. The condition is in general indispensable in our theorem. For example, the unit groups of all maximal orders in certain definite quaternion algebras over ko consist of the units ± 1 only. The proof of the theorem will be based on the following Hilfssatz due to Eichler [3, p. 239, Hilfssatz 9].