On two classical theorems in the theory of orders (Q750571)

Let A be a central simple algebra over a global field K. A ``bouquet'' of orders in A is a collection consisting of all the maximal orders of A that have (i) the same intersection S with a given maximal commutative subfield of A, and (ii) the same completion at each ramification point of A. The two theorems of the title are as follows: The Chevalley-Hasse-Noether theorem: Suppose S is a maximal order. Then for \(\Lambda\) and \(\Lambda '\) two maximal orders in the bouquet defined by S, there is an S-ideal \({\mathfrak a}\) with \(\Lambda '{\mathfrak a}={\mathfrak a}\Lambda\). - The Latimer- MacDuffee theorem: Let \(S={\mathbb{Z}}[\vartheta]\), where \(\vartheta\) is the root of a monic separable irreducible integer polynomial of degree n. Then there is a bijection between the ideal classes of S and the \(GL_ n({\mathbb{Z}})\)-orbits of ring embeddings of S in \(\Lambda\) (under conjugation). - The author shows that these results are closely related in that they are ``dual'' to one another in an appropriate setting in which the idèle group of A acts both on the set of embeddings of S in \(\Lambda\) and on the set of maximal orders in a bouquet. This enables him to generalize both theorems, and to find some relations between the class numbers of the various orders.