Maximal orders in rational cyclic algebras of composite degree

Introduction. A maximal order M of a normal division algebra D over the rational number field may be imbeddedt in a simple fashion in a maximal order of any normal simple algebra similar to D. When the normal simple algebra has degree greater than two, its class number is unity,j and it can then be shown that all maximal orders of the algebra are obtainable from any one by an inner automorphism of the algebra. Thus it is sufficient to determine a single M of each D in order to determine all maximal orders of all normal simple algebras of degree greater than two over the rational number field. This determination was made by Hull? for the case in which the degree n of D is any odd prime, using methods similar to those of Albertff for the case n=2. The methods and results of Hull are extended here to the case in which n = 7re where 7r is any odd prime, and also to the case n = 2e >2 provided that D has odd discriminant and has the real number field as splitting field. More specifically, it will be shown with the aid of the class field theory that each algebra D considered has a suitably normalized cyclic generation, and a maximal order of D will be expressed in terms of a finite number of quantities related to this generation. There are two chief points of difference between the present case and that of prime degree. The quantity ain the normalized generation (Z, S, v-) is no longer the product of the primes ramified in D, but the product of certain powers of these primes. The exponents on these powers reduce to unity in the case of prime degree. The explicit basis given for the maximal order is similar to that for prime degree